GEOMETRY OF THE COMPLEX OF CURVES I: HYPERBOLICITY

HOWARD A. MASUR AND YAIR N. MINSKY

May 13, 1996

1. Introduction

In topology, geometry and complex analysis, it is useful to attach a number of interesting mathematical objects to a surface S. The Teichmüller space T (S) is the parameter space of conformal (or hyperbolic) structures on S, up to isomorphism isotopic to the identity. The Mapping Class Group Mod(S) is the group of autohomeomorphisms of S, up to isotopy. The geometric and group-theoretic properties of these objects are tied to each other via the intrinsic combinatorics, or topology, of S.

In [15], Harvey associated to a surface S a finite-dimensional simplicial complex C(S) , called the complex of curves, which was intended to capture some of this combinatorial structure, and in particular to encode the asymptotic geometry of Teichmüller space in analogy with Tits buildings for symmetric spaces. The vertices of Harvey’s complex are homotopy classes of simple closed curves in S, and the simplices are collections of curves that can be realized disjointly. This complex was then considered by Harer [13, 14] from a cohomological point of view and by Ivanov [18, 17, 19] with applications to the structure of Mod(S) (in particular a new proof of Royden’s theorem).

In this paper we begin a study of the intrinsic geometry of C(S), which can be made into a metric space in a standard way by making each simplex a regular Euclidean simplex of sidelength 1. Our main result is the following:

Theorem 1.1. (Hyperbolicity) Let S be an oriented surface of finite type. The curve complex C(S) is a δ-hyperbolic metric space, where δ depends on S. Except when S is a sphere with 3 or fewer punctures, C(S) has infinite diameter.

(See §2.1 for a definition of δ-hyperbolicity.)

We remark that in a few sporadic cases our definition of C(S) varies slightly from the original; see §2.2. Note also that we can just as well consider the 1-skeleton C_{1}(S) rather than the whole complex: δ-hyperbolicity is a quasi-isometry invariant, and C(S) is evidently quasi-isometric to its 1-skeleton.

The first author was partially supported by NSF grant #DMS 9201321. The second author was partially supported by an NSF postdoctoral fellowship and a fellowship from the Alfred P. Sloan Foundation.

This theorem is motivated in part by the need to understand the extent of an important but incomplete analogy between the geometry of the Teichmüller space and that of complete, negatively curved manifolds, or more generally of δ-hyperbolic spaces. There are many senses in which this analogy holds, and it was exploited, for example, by Bers [2], Kerckhoff [22], and Wolpert [38]. On the other hand, Masur [25] showed that (except for the simplest cases) the Teichmüller metric on T (S) cannot be negatively curved in a local sense, and more recently Masur-Wolf [29] showed that it is not δ-hyperbolic. The Weil-Petersson metric on T (S) has negative sectional curvatures, however they are not bounded away from zero [26, 39].

The failure of δ-hyperbolicity in T (S) is closely related to the presence of infinite diameter regions where the metric on T (S) is nearly a product (let us consider from now on only the Teichmüller metric on T (S)). Fixing a small ϵ_{0} > 0, let

H_{α} = {x ∈ T (S) : Ext_{x}(α) ≤ ϵ_{0}}

denote the region in T (S) where a simple closed curve α has small extremal length (see Section 2 for definitions). Then (see Minsky [30]) the Teichmüller metric in this region is approximated by a product of infinite-diameter metric spaces, and so cannot be δ-hyperbolic.

As a consequence of the Collar Lemma (see e.g. [20, 4]), when ϵ_{0} is sufficiently small the intersection pattern of the family {H_{α}} is exactly encoded by the complex C(S) (it is the nerve of this family). Thus, one interpretation of our main theorem is that the regions {H_{α}} are the only obstructions to hyperbolicity, and once their internal structure is ignored, the way in which they fit together is hyperbolic.

This can be made precise by Farb’s notion of relative hyperbolicity [8], and in Section 7 we will prove:

Theorem 1.2. (Relative Hyperbolicity 1) The Teichmüller space T (S) is relatively hyperbolic with respect to the family of regions {H_{α}}.

A similar discussion can be carried out for the mapping class group. A group is word hyperbolic if its Cayley graph is δ-hyperbolic. It is known that a group acting by isometries on a δ-hyperbolic space with finite point stabilizers and compact quotient must itself be word-hyperbolic, and it is plain that Mod(S) acts isometrically on C(S), with compact quotient. Nevertheless, Mod(S) is known not to be wordhyperbolic for all but the simplest cases, because it contains high-rank abelian subgroups. This is not a contradiction, because the action on C(S) has infinite point stabilizers.

Indeed, abelian subgroups in Mod(S) are generated by elements that stabilize disjoint subsurfaces, and in particular their boundary curves, and hence are “invisible” from the point of view of coarse geometry of the complex. One can formalize this intuition as we did with Teichmüller space by considering subgroups of Mod(S) fixing certain curves, and their cosets. In Section 7 we will carry this out and prove:

Theorem 1.3. (Relative Hyperbolicity 2) The group Mod(S) is relatively hyperbolic with respect to left-cosets of a finite collection of stabilizers of curves.

Farb shows in [8] that relative-hyperbolicity results such as these are useful in converting information about subgroups (such as automaticity) to information about the full groups. Although his work does not apply directly to our situation, it is nonetheless possible to use the results of this paper as the first step in an inductive analysis of the structure of the Mapping Class Group. We hope to carry this out in a future paper [24].

We remark finally that although our main theorem has essentially a topological statement, the proof we have found uses Teichmüller geometry in an essential way. It would be very interesting to find a purely combinatorial proof. In particular, it would be nice to have an effective estimate of the constant δ, which our proof does not provide since it depends on bounds obtained from a compactness argument in the Moduli space.

2. Outline of the Proof

In this section, after describing some necessary background, we will give an outline of the proof of the Hyperbolicity Theorem 1.1, which reduces it to a number of assertions. These assertions will then be proven in sections 3 through 6.

2.1. Hyperbolicity. A geodesic metric space X is a path-connected metric space in which any two points x, y are connected by an isometric image of an interval in the real line, called a geodesic and denoted [xy] (we use this notation although [xy] is not required to be unique).

We say that X satisfies the thin triangles condition if there exists some δ ≥ 0 such that, for any x, y and z ∈ X the geodesic [xz] is contained in a δ-neighborhood of [xy] ∪ [yz]. This is one of several equivalent conditions for X to be δ-hyperbolic in the sense of Gromov, or negatively curved in the sense of Cannon. (We remark that there are formulations of hyperbolicity that do not require X to be a geodesic space, but we will not be concerned with them here. See Cannon [5], Gromov [12] and also [3, 7, 11].)

Important examples of hyperbolic spaces are the classical hyperbolic space H^{n}, all simplicial trees (here δ = 0), and Cayley graphs of fundamental groups of closed negatively curved manifolds.

We note also that every finite-diameter space is trivially δ-hyperbolic with δ equal to the diameter, which is the reason we must check that the complex of curves has infinite diameter.

2.2. The complex of curves. Let S be a closed surface of genus g with p punctures. Except in the sporadic cases mentioned below, define a complex C(S) as follows: k-simplices of C(S) are (k + 1)-tuples {γ_{0}, γ_{1}, …, γ_{k}} of distinct non-trivial homotopy classes of simple, non-peripheral closed curves, which can be realized disjointly. This complex is obviously finite-dimensional by an Euler characteristic argument, and is typically locally infinite.

Sporadic cases. In a number of cases C(S) (and hence our main theorem) is either trivial or already well-understood. When S is a sphere (g = 0) with p ≤ 3 punctures, the complex is empty. In this case we can say Theorem 1.1 holds vacuously. When g = 0 and p = 4, or g = 1 and p ≤ 1, Harvey’s complex has no edges, and is just an infinite set of vertices. In these cases it is useful to alter the definition slightly, so that edges are placed between vertices corresponding to curves of smallest possible intersection number (1 for the tori, 2 for the sphere). When this is done, we obtain the familiar Farey graph, for which Theorem 1.1 is fairly easy to prove. See [31] for an exposition of this case.

For the remainder of the paper we exclude the surfaces with g = 0, p ≤ 4 and g = 1, p ≤ 1, which we call sporadic.

In all other cases, the dimension of the complex is easily computed to be 3g−4+p, which in particular is at least 1. Letting C_{k} denote the k-skeleton of C, we focus on the graph C_{1}. We turn C_{1} into a metric space by specifying that each edge has length 1, and we denote by d_{C} the distance function obtained by taking shortest paths. Note also that C_{1} is a geodesic metric space.

Let i(α, β) denote the geometric intersection number of α with β, which is equal to the number of transverse intersections of their geodesic representatives in a hyperbolic metric on S.

Lemma 2.1. If S is not sporadic, C_{1} is connected. Moreover for any two curves α, β, d_{C}(α, β) ≤ 2i(α, β) + 1.

Remark. In fact for large d_{C} a better estimate is that i(α, β) is at least exponential in d_{C}, as we shall see in Section 3.

Proof. Assume that α and β are realized with minimal intersection number If i(α, β) = 1 then a regular neighborhood of α ∪ β is a punctured torus whose boundary γ must be nontrivial and nonperipheral since the torus and punctured torus are excluded. Since γ is disjoint from both α and β, d(α, β) = 2.

For i(α, β) ≥ 2, fixing two points of α ∩ β adjacent in α there are two distinct ways to do surgery on these points, replacing a segment of β with the segment of α between them, producing homotopically nontrivial simple curves β_{1}, β_{2} such that i(α, β_{j}) ≤ i(α, β) − 1. If the two intersections agree in orientation then i(β, β_{j}) = 1, and neither β_{j} can be peripheral (if it bounds a punctured disk then α must enter it and has a non-essential intersection with β). Thus d(α, β) ≤ 2 + d(α, β_{j}) and we are done by induction.

If the two intersections have opposite signs then actually i(α, β_{j}) ≤ i(α, β) − 2, and i(β, β_{j}) = 0 for j = 1, 2. Thus if at least one β_{j} is nonperipheral, we again apply induction (and get a better estimate than above). If both β_{1}, β_{2} are peripheral then β must bound a twice punctured disk on the side containing the α segment between the intersections. Thus consider a segment of α between intersections, which is adjacent to this one. If it also falls into the last category then β bounds a twice punctured disk on its other side too, and S must be a 4-times punctured sphere, which has been excluded.

2.3. Teichmüller space. An analytically finite conformal structure on S is an identification of S with a closed Riemann surface minus a finite number of points. Let T (S) denote the Teichmüller space of analytically finite conformal structures on S, modulo conformal isomorphism isotopic to the identity.

Given an element x ∈ T (S) and a simple closed curve α in S, we recall that the extremal length Ext_{x}(α) is the reciprocal of the largest conformal modulus of an embedded annulus in S homotopic to α. We remark also that an alternate definition is Ext_{x}(α) = sup_{σ} |α^{∗}|_{σ}^{2} where σ ranges over conformal metrics of area 1 on (S, x), and |α^{∗}|_{σ} denotes σ-length of a shortest representative of α. (See e.g. Ahlfors [1].)

The Teichmüller metric d_{T} on T (S) can be defined in terms of maps with minimal quasiconformal dilatation, but for us it will be useful to note Kerckhoff’s characterization [21]:

d_{T} (x, y) = \frac{1}{2} log sup_{α∈C_{0}(S)} \frac{Ext_{y}(α)}{Ext_{x}(α)}. (2.1)

A holomorphic quadratic differential q on a Riemann surface is a tensor of the form φ(z)dz^{2} in local coordinates, with φ holomorphic. Away from zeroes, a coordinate ζ can be chosen so that q = dζ^{2}, which determines a Euclidean metric |dζ^{2}| together with a pair of orthogonal foliations parallel to the real and imaginary axes in the ζ plane. These are well-defined globally and are called the horizontal and vertical foliations, respectively. (See Gardiner [10] or Strebel [35].)

Geodesics in T (S) are determined by quadratic differentials. Given q holomorphic for some x ∈ T (S), for any t ∈ R we consider the conformal structures obtained by scaling the horizontal foliation of q by a factor of e^{t}, and the vertical by e^{−t}. The resulting family, which we write L_{q}(t), is a geodesic parametrized by arclength.

For a closed curve or arc α in S, denote by |α|_{q} its length in the q metric. Let |α|_{q,h} and |α|_{q,v} denote its horizontal and vertical lengths, respectively, by which we mean the total lengths of the (locally defined) projections of α to the horizontal and vertical directions of q.

Finally we note that the variation of horizontal and vertical lengths is given by

|α|_{q_{t},h} = |α|_{q_{0},h}e^{t} (2.2)

and

|β|_{q_{t},v} = |β|_{q_{0},v}e^{−t}. (2.3)

2.4. The proof of the Hyperbolicity Theorem. One way to prove hyperbolicity is to find a class of paths with the following contraction property:

Definition 2.2. Let X be a metric space. Say that a path γ : I → X (where I ⊂ R is some interval, possibly infinite) has the contraction property if there exists π : X → I and constants a, b, c > 0 such that:

(1) For any t ∈ I, diam(γ([t, π(γ(t))])) ≤ c

(2) If d(x, y) ≤ 1 then diam γ([π(x), π(y)]) ≤ c.

(3) If d(x, γ(π(x))) ≥ a and d(x, y) ≤ bd(x, γ(π(x))), then

diam γ[π(x), π(y)] ≤ c.

(Here for s, t ∈ R we take [s, t] to mean the interval with endpoints s, t regardless of order.)

One should think of this property in analogy with closest-point projection to a geodesic in H^{n}. Condition (1) is a coarsening of the requirement that points in γ(I) be fixed. Condition (2) states that the projection is coarsely Lipschitz. Condition (3) is the most important, stating that the map is, in the large, strongly contracting for points far away from their images in γ(I). Note that this holds in H^{n} for b = 1.

Note also that we give π as a map to the parameter interval I rather than its image, in order to avoid requiring anything about the speed of parametrization of γ: for example γ is allowed to be constant for long intervals, and on the other hand it need not be continuous.

We say that a family Γ of paths has the contraction property if every γ ∈ Γ has the contraction property, with respect to a uniform a, b, c > 0.

Call a family of paths coarsely transitive if there exists D ≥ 0 such that for any x and y with d(x, y) ≥ D there is a path in the family joining x to y. In section 6 we will prove the following theorem, which is probably well-known.

Theorem 2.3. If a geodesic metric space X has a coarsely transitive path family Γ with the contraction property then X is hyperbolic. Furthermore, the paths in Γ are quasi-geodesics.

(See §6 for the definition of quasi-geodesic in this context).

Our family of paths will be constructed using Teichmüller geodesics, in the following manner. There is a natural map Φ from T (S) to finite subsets of C(S), assigning to any x ∈ T (S) the set of curves of shortest Ext_{x} (extremal length is convenient for us, though hyperbolic will do as well). A geodesic in T (S) traces out, via Φ, a path in C(S) up to some bounded ambiguity.

That is, let q be a quadratic differential on a Riemann surface x and let L_{q} : R → T (S) be the corresponding Teichmüller geodesic (parametrized by arclength). Let a map

F ≡ F_{q} : R → C(S)

be defined by assigning to t one of the curves of Φ(L_{q}(t)). The actual choices will not matter, as Φ(x) has uniformly bounded diameter:

Lemma 2.4. There exists c = c(S) such that diam_{C} Φ(x) ≤ c for all x ∈ T (S).

Proof. There exists e_{0}(S) such that the shortest nonperipheral curve on (S, x) has extremal length at most e_{0}. Thus Lemma 2.5 below immediately bounds the distance between any two shortest curves, by 2e_{0} + 1.

Note in fact that there exists ϵ_{0} such that if x has a curve α of extremal length at most ϵ_{0} then any curve intersecting α has extremal length greater than ϵ_{0}. In this case the diameter of Φ(x) is at most 1.

Lemma 2.5. For α, β ∈ C_{0}(S), if Ext_{x}(α) ≤ E and Ext_{x}(β) ≤ E for some conformal structure x on S, then d_{C}(α, β) ≤ 2E + 1.

Proof. It is an elementary fact (see e.g. [33]) that Ext_{x}(α)Ext_{x}(β) ≥ i(α, β)^{2}. Thus the assumption of the lemma gives i(α, β) ≤ E. Now by Lemma 2.1, d_{C}(α, β) ≤ 2E + 1.

If q has a closed vertical leaf then there is a collection of (up to homotopy) disjoint vertical curves whose extremal lengths go to 0 as t → ∞. In this case choose a fixed one of these to be the value of F as t → ∞, and let this also be denoted by F (∞). Similarly define F (−∞) if there are horizontal curves.

The projection for F will be defined using the notion of balance. Recalling the notation of §2.3, we say that β is balanced with respect to q if |β^{∗}|_{q,h} = |β^{∗}|_{q,v}, where β^{∗} is a q-geodesic representative (it may be necessary for β^{∗} to go through punctures – see §4.1).

We note that β^{∗} is also geodesic with respect to any q_{t}. Since | · |_{q_{t},h} and | · |_{q_{t},v} vary like e^{t} and e^{−t} (by (2.2) and (2.3)), if β^{∗} is not entirely vertical or horizontal with respect to q there is a unique t for which β is balanced, and this is also the minimum of the quantity |β^{∗}|_{q_{t},h} + |β^{∗}|_{q_{t},v}. We observe also that, since the q-length of β^{∗} is estimated by

\frac{1}{\sqrt{2}}(|β^{∗}|_{q,h} + |β^{∗}|_{q,v}) ≤ |β^{∗}|_{q} ≤ |β^{∗}|_{q,h} + |β^{∗}|_{q,v},

the minimum of |β^{∗}|_{q_{t}} also occurs within bounded distance (in fact \frac{1}{2} cosh^{−1} \sqrt{2}) of the balance point. (Compare with the projection used in [34]).

Let C_{b} = C_{b}(q) denote the set of simple closed curves that are not entirely horizontal or vertical for q. We define π = π_{q} : C_{0} → R as follows: for β ∈ C_{b} let π(β) be the unique t for which β is balanced for q_{t}. For β ∈ C \ C_{b} let π(β) be +∞ if β is vertical, and −∞ if β is horizontal. (As above, in this case F (±∞) makes sense).

Suppose now d(α, β) ≥ 3. Then α and β fill S, in that there is no γ disjoint from both. There is therefore a quadratic differential q whose nonsingular vertical leaves are homotopic to α and whose nonsingular horizontal leaves are homotopic to β. Then F_{q}(+∞) = α and F_{q}(−∞) = β. This shows that the family {F_{q}} is coarsely transitive.

Hyperbolicity will therefore be a consequence of Theorem 2.3 and the following:

Theorem 2.6. (Projection Theorem) The path family {F_{q}} satisfies the contraction property with the projections π_{q} defined above.

The proof of this theorem will be given in section 5.

We will begin in Section 3 by developing tools for controlling distances between curves in C_{0}(S). Using Thurston’s train-track coordinates, we will analyze a covering of C_{0}(S) by a family of polyhedra which have the property that a point contained in a deeply nested sequence of polyhedra will be a definite distance from any point outside the outermost polyhedron (Lemma 3.2). A partial converse to this will be the Nesting Lemma 3.7, which given two distant curves will allow us to construct a nested sequence of polyhedra separating them. We will apply these tools to prove Lemma 3.12, which relates intersection numbers to distance in C(S) in a way which can be directly applied in Section 5.

Proposition 3.6 in Section 3.3 will establish the infinite-diameter claim in the Hyperbolicity Theorem 1.1.

3. The nested train-track argument

3.1. Train-tracks. We refer to Penner-Harer [37] for a thorough treatment of train-tracks, recalling here some of the terminology. A train-track on a surface S is an embedded 1-complex τ satisfying the following properties. Each edge (called a branch) is a smooth path with well-defined tangent vectors at the endpoints, and at any vertex (called a switch) the incident edges are mutually tangent. The tangent vector at the switch pointing toward the interior of an edge can have two possible directions, and this divides the ends of edges at the switch into two sets, neither of which is permitted to be empty. Call them “incoming” and “outgoing”. The valence of each switch is at least 3, except possibly for one bivalent switch in a closed curve component. Finally, we require that the components of S \ τ have negative generalized Euler characteristic, in this sense: A surface R whose boundary consists of smooth arcs meeting at cusps has a generalized Euler characteristic which is the Euler characteristic of R minus 1/2 for every outward-pointing cusp (internal angle 0), plus 1/2 for every inward-pointing cusp (internal angle 2π). For the train track complementary regions all cusps are outward, so that we exclude annuli, oncepunctured disks with smooth boundary, or unpunctured disks with 0, 1 or 2 cusps at the boundary. We will usually consider isotopic train-tracks to be the same.

A train route is a non-degenerate smooth path in τ; in particular it traverses a switch only by passing from incoming to outgoing edge (or vice versa). A transverse measure on τ is a non-negative function μ on the branches satisfying the switch condition: For any switch the sums of μ over incoming and outgoing branches are equal. A closed train-route induces the counting measure on τ.

A train-track is recurrent if every branch is contained in a closed train route, or equivalently if there is a transverse measure which is positive on every branch.

Let ML(S) denote the space of measured geodesic laminations on S (see e.g. [22, 16]). A geodesic lamination λ is carried on τ if there is a homotopy of S taking λ to a set of train routes. In such a case λ induces a transverse measure on τ, which uniquely determines it. The set of measures on τ gives local coordinates on ML(S), and in fact ML(S) is a manifold (homeomorphic to a Euclidean space).

For a recurrent train-track τ, let P (τ) denote the polyhedron of measures supported on τ. We will blur the distinction between P (τ) as a subset of ML(S), and as a subset of the space R_{+}^{B} of non-negative functions on the branch set B of τ. The dimension dim(P (τ)) of the polyhedron will also be written dim(τ).

We note that P (τ) is preserved by scaling, so it is a cone on a compact polyhedron in projective space. However we will need to consider actual measures in P (τ) rather than projective classes.

By int(P (σ)) we will mean the set of weights on σ which are positive on every branch. Note that unless dim(σ) is maximal, this is different from the interior of P (σ) as a subset of ML(S), which is empty.

We write σ < τ if σ is a subtrack of τ; that is, σ is a train track which is a subset of τ. We also say that τ is an extension of σ in this case. We write σ ≺ τ if σ is carried on τ, by which we mean that there is a homotopy of S such that every train route on σ is taken to a train route on τ. It is easy to see that σ < τ is equivalent to P (σ) being a subface of P (τ), and σ ≺ τ is equivalent to P (σ) ⊆ P (τ).

Say that σ fills τ if σ ≺ τ and int(P (σ)) ⊆ int(P (τ)). When both tracks are recurrent this is equivalent to saying that every branch of τ is traversed by some branch of σ. Similarly, a curve α fills τ if α ≺ τ and traverses every branch of σ.

Call a train-track τ large if all the components of S \ τ are polygons or oncepunctured polygons. We will also say that P (τ) is large in such a situation.

A vertex cycle of τ is a positive measure on τ which is an extreme point of P (τ). That is, its projective class is a vertex of the projectivized polyhedron. Up to scaling, a vertex cycle can always be realized by the counting measure on a single, simple closed curve, and we will always assume that a vertex is of this form.

Splitting. Let τ be a generic train-track (all switches are trivalent). A splitting move is one of the three elementary moves on a local configuration, as shown in figure 1. The three splits are called a left split, a collision, and a right split, and the resulting tracks in each case are carried by τ. If we place a positive measure μ on τ and use the labelling as in the figure, we note that a positive measure is induced on the right split track if μ(a) > μ(c), on the left split track if μ(a) < μ(c), and on the collision track if μ(a) = μ(c). We call a a winner of the splitting operation if μ(a) > μ(c) (note that d is then also a winner).

Figure 1. The three ways to split through an edge.

Any measured lamination β carried on τ determines a sequence of possible splittings by the rule in the previous paragraph, and all the resulting train-tracks carry β. Note that if β fills τ then it will continue to fill the split tracks, and in particular they will all be recurrent. This process can continue as long as the split track is not a simple closed curve. (We must check that for any recurrent track that is not a simple curve there is a “splittable” edge, that is one which is in the configuration of figure 1: simply consider any transverse measure whose support is all of τ and take an edge of maximal weight. See [23]).

When β is a simple closed curve we will usually terminate the sequence as soon as we reach a track for which β is a vertex.

Note also that if σ is a right or left splitting of τ then P (σ) and P (τ) must share at least one vertex. To see this, note that P (σ) is one of the pieces obtained by cutting P (τ) by a hyperplane. Such a subset always contains a vertex of the original polyhedron. In the case of a collision splitting, we at least see that σ is a subtrack of a track that shares a vertex with τ.

Finally, we note that when τ is not generic, each switch can be slightly perturbed (“combing” in [37]) to yield a generic track carried by τ which carries the same set of laminations.

Diagonal extensions. Let σ be a large track. A diagonal extension of σ is a track κ such that σ < κ and every branch of κ \ σ is a diagonal of σ: that is, its endpoints terminate in corner of a complementary region of σ. Let E(σ) denote the set of all recurrent diagonal extensions of σ. Note that it is a finite set, and let P E(σ) denote ⋃_{κ∈E(σ)} P (κ).

Further, let us define N(τ) to be the union of E(σ) over all large subtracks σ < τ. Define P N(τ) = ⋃_{κ∈N(τ)} P (κ). In some sense this should be thought of as a “neighborhood” of P (τ); compare Lemma 3.4 and (3.1).

Let int(P E(σ)) denote the set of measures μ ∈ P E(σ) which are positive on every branch of σ. We also define int(P N(τ)) = ⋃_{κ} int(P E(κ)), where κ varies over the large subtracks of τ.

The following is a sufficient condition for containment in int(P E(σ)).

Lemma 3.1. There exists δ > 0 (depending only on S) for which the following holds. Let σ < τ where σ is a large track. If μ ∈ P (τ) and, for every branch b of τ \ σ and b′ of σ, μ(b) < δμ(b′), then σ is recurrent and μ ∈ int(P E(σ)).

Proof. Whenever there are branches of τ \ σ which meet an edge e of the boundary of a complementary region of σ at other than a corner point, there is a splitting move involving these branches which replaces σ by an equivalent track (also called σ) and either reduces the number of edges of τ \ σ incident to σ, or moves one of them closer to a corner (see figure 2).

As soon as a branch of τ \ σ is separated from a corner of a complementary region only by non-splittable edges, and facing the right way, we can perform a shift move (see figure 3) which takes this branch to the corner without affecting the set of measures carried on the track. Thus a bounded number of such splitting and shifting moves takes τ to a track τ′ ∈ E(σ).

In order for the measure μ to be carried on τ′, it must be consistent with the sequence of splittings. That is, whenever a splitting is determined by a comparison between a branch b of τ \σ and a branch c of σ, the branch of τ \σ must lose (that is, μ(b) < μ(c)). After such a splitting, there is a branch of σ with measure μ(c)−μ(b).

Figure 2. Solid edges are in σ, dotted edges in τ \ σ. In (a), the splitting reduces the number of dotted edges incident to σ. In (b), the splitting brings a dotted edge closer to a corner.

Figure 3. Shifting an edge of τ \ σ into a corner

Thus, if k is the bound on the number of splittings that take place, a sufficient condition that all the splittings are consistent with μ is min_{σ} μ > (k + 1) max_{τ\σ} μ.

Therefore, setting δ = 1/(k + 1), we guarantee that μ ∈ P E(σ), and furthermore that μ puts positive measure on each branch of σ after the splitting, so that μ ∈ int(P E(σ)).

Finally, we must show that σ is recurrent. It is an easy fact of linear algebra that, if there is an assignment of positive weights on the branches of σ such that at every switch the difference of incoming and outgoing weights is less than a fixed constant δ_{1} times the minimum weight (where δ_{1} depends just on the surface S), then these weights can be perturbed to positive weights satisfying the switch conditions, and hence σ is recurrent. It follows that, with sufficiently small δ, the measure μ restricted to σ gives such a set of weights.

The next two lemmas show that, when tracks are nested, their diagonal extensions are nested in a suitable sense, and the way in which the diagonal branches cover each other is controlled.

Lemma 3.2. Let σ and τ be large recurrent tracks, and suppose σ ≺ τ. If σ fills τ, then P E(σ) ⊆ P E(τ). Even if σ does not fill τ, we have P N(σ) ⊆ P N(τ).

Proof. We may thicken τ slightly to get a regular neighborhood τ_{ϵ}, which can be foliated by short arcs called “ties” transverse to τ. Then σ can be embedded in τ_{ϵ} so that it is transverse to the ties.

The assumption that σ fills τ implies that every edge of τ is traversed by some edge of σ, and thus σ crosses every tie. Any component D of S \ σ, which is a possibly once-punctured polygon, must have some subset F foliated by ties. F consists of a neighborhood of the boundary and bands joining different boundary edges. Each component of D\F is isotopic to a component of S\τ, and the quotient of D obtained by identifying each tie to a point can be identified with some union of complementary regions of τ, joined by train routes in τ. Any diagonal edge e in D joining two corners of D may therefore be put in minimal position with respect to the ties (so that e meets ties transversely, and no disks are bounded by a segment of e and a tie), and hence gives rise to a train route through the union of τ with some diagonal edges.

It follows that any diagonal extension of σ can be carried by a diagonal extension of τ, and hence P E(σ) ⊂ P E(τ).

Now in general, if κ is a large subtrack of σ, let ρ be the smallest subtrack of τ carrying κ. Note that ρ is necessarily large, and κ fills ρ. Thus the same argument applies to the faces, and we can conclude P N(σ) ⊆ P N(τ)

Lemma 3.3. Let σ ≺ τ where σ is a large recurrent track, and let σ′ ∈ E(σ), τ′ ∈ E(τ) such that σ′ ≺ τ′. Then any branch b of τ′ \ τ is traversed with bounded degree m_{0} by branches of σ′. The number m_{0} depends only on S.

Proof. It will suffice to show that no branch of σ′ passes through a branch of τ′ \ τ more than twice; we then obtain m_{0} from a topological bound on the number of branches of any train track in S.

As in the previous lemma, let τ′_{ϵ} be a regular neighborhood of τ′ foliated by ties and isotope σ′ so that it is contained in τ′_{ϵ} and is transverse to the ties. Because each component of S \ τ′ is either a polygon with d ≥ 3 corners or a once-punctured polygon with p ≥ 1 corners, we may extend the ties to a foliation F of S with one index 1 − d/2 ≤ −1/2 singularity in each d-gon and an index 1 − p/2 ≤ 1/2 singularity at the puncture of each punctured p-gon. (See figure 4.)

Figure 4. The foliation F in a triangular component, and in a punctured monogon.

Fix a tie t that crosses the regular neighborhood of a branch b of τ′ \ τ. Since σ ≺ τ, no branch of σ crosses t. Suppose a branch e of σ′ \ σ crosses t twice and let t′ be a segment in t between two successive crossings. Then t′ and an interval e′ ⊂ e form an embedded loop in a complementary region U of σ, which is either a disk or once-punctured disk since σ is large.

Since the foliation is transverse to e and parallel to t, we can see that the index of the foliation around t′ ∪ e′ is therefore +1 if the ends of e′ meet t on opposite sides, and +1/2 if they meet t on the same side. The first case cannot occur since the disk bounded by t′ ∪ e′ can contain at most one singularity of F, of index at most 1/2. The second case can occur if t′ ∪ e′ surrounds a puncture. However, in this case we can see that there are no other points of t ∩ e. For if there were we could combine two such loops to find a loop in U with index 1, again a contradiction.

3.2. Nesting and C-distance. Although it is relatively easy to understand geometrically when pairs of curves are a distance at most 2 in C_{1}, larger distances are more subtle to detect. Begin with the observation that, if α and β are disjoint curves and α is carried on a maximal train-track σ (one all of whose complementary regions are triangles or punctured monogons) in such a way that it passes through every branch, then β is also carried on σ. This is a special case of the following more general fact:

Lemma 3.4. If σ is a large recurrent train-track and α ∈ int(P E(σ)) then

d_{C}(α, β) ≤ 1 ⟹ β ∈ P E(σ).

In other words,

N_{1}(int(P E(σ))) ⊂ P E(σ),

where N_{1} denotes a radius 1 neighborhood in C_{1}.

Proof. Let τ ∈ E(σ) be the support of α. If we assume that β ≠ P E(τ), then the more quantitative Lemma 3.5, below, applies to give in particular that i(α, β) > 0. Since P E(τ) ⊂ P E(σ), we are done.

We note the following immediate consequence:

N_{1}(int(P N(σ)) ⊂ P N(σ) (3.1)

which is obtained by applying Lemma 3.4 to the large subtracks of σ.

It remains to prove the following lemma, which will also be used at the end of this section.

Lemma 3.5. Let τ be a large recurrent track, let α be carried on a diagonal extension τ′ ∈ E(τ), and let β be a curve not carried on any diagonal extension of τ. Then i(α, β) ≥ min_{b} α(b) where the right hand side denotes the minimum weight α puts on all branches b of τ.

Proof. Consider first the case that τ′ = τ, so that α is actually carried in τ.

Fix a hyperbolic metric on S, and lift τ to a train-track τ̃ in the universal cover H^{2}. The generalized Euler characteristic defined in §3.1 is additive for unions of (closures of) complementary regions of τ, and one can show (see Casson-Bleiler [6]) that this together with the fact that there are finitely many isometry types of regions, implies that each train route r of τ̃ is uniformly quasi-geodesic, and in particular has two distinct endpoints ∂r = {r_{+}, r_{−}} on the circle ∂H^{2} and stays in a uniform neighborhood of the geodesic r^{∗} connecting them.

Choose a component β̃ of the lift of β to H^{2}, and note it is also quasi-geodesic. Let T_{β} be a generator of the subgroup of π_{1}(S) preserving β̃. We say that an edge ẽ of τ̃ separates β̃ consistently if, for any train route r passing through ẽ, its endpoints r_{±} separate β̃_{±}. If this occurs for some ẽ then, letting e be its projection to S, we deduce immediately that i(β, α) ≥ α(e), since α lifts to a collection of train routes with α(e) of them passing through ẽ, and through each translate T_{β}^{m}(ẽ).

Thus, let us now prove that, if no edge of τ̃ separates β̃ consistently, then β̃ is carried on a diagonal extension of τ̃ (at the end we will check that this projects down to an extension of τ).

Each train route r separates H^{2} into two open disks, which we call halfplanes. Note that each is contained in a uniformly bounded neighborhood of a geodesic halfplane bounded by r^{∗}. For a halfplane H, let H′ denote H̄ \ r̄ (where the bar denotes closure in the closed disk), i.e. the union of H with an open arc on the boundary circle.

Let J_{+} and J_{−} be the components of ∂H^{2} \ ∂β̃. Let H_{+} denote the union of all halfplanes H (bounded by train routes) such that H′ meets the boundary entirely in J_{+}, and define H_{−} similarly.

Note that each H_{±} is open and contained in a bounded neighborhood of a geodesic halfplane bounded by β̃^{∗}, and furthermore that H_{−} and H_{+} are disjoint: For if not, there would be two halfplanes H_{+}, H_{−} meeting only J_{+} and J_{−}, respectively, at infinity, whose intersection is nonempty and hence must be a bigon bounded by arcs of their train route boundaries. This contradicts the Euler characteristic condition on τ̃.

Let K = H^{2} \ (H_{+} ∪ H_{−}). This is a closed set bounded by pieces of train routes, and we observe it is also connected. To see this, we think of K as the intersection of a sequence of closed half-planes C_{j}, and show that it is connected at any finite stage: given a connected finite intersection K_{N} = ⋂_{j≤N} C_{j}, the train route boundary r_{N+1} of C_{N+1} must intersect K_{N} in a connected arc, for otherwise we obtain an arc in H^{2} \ K_{N} with endpoints on K_{N}, which together with a piece of ∂K_{N} must bound a region of non-negative Euler characteristic, again a contradiction. It follows that K_{N} ∩ C_{N+1} is connected. A decreasing sequence of connected sets is connected, hence K is.

We also claim that its interior is disjoint from τ̃. For, if ẽ is any branch of τ̃, by our assumption ẽ does not consistently separate β̃, and hence lies on a train route both of whose endpoints are in either J̄_{+} or J̄_{−}. Thus, ẽ is on the boundary of one of the halfplanes comprising either H_{+} or H_{−}, and hence in H̄_{±}. (Note, this is the only place where we use the assumption).

Thus every component P of int(K) is a polygonal component of H^{2} \ τ̃, possibly infinite-sided.

The complement in K of the closure of int(K) must be a union of arcs which are train routes, and every such arc a must lie on the common boundary of two disjoint half-planes H_{a}^{+}, H_{a}^{−}. In particular ā meets the closure of a polygonal region P ⊂ int(K) only at a cusp point. We claim that the component of H^{2} \ (H_{a}^{+} ∪ H_{a}^{−}) containing P must meet an endpoint of β̃ at infinity: this component, adjoined to H_{a}^{+}, yields a half-plane bounded by a train route traversing half of ∂H_{a}^{+} and half of ∂H_{a}^{−}, and if the region fails to meet an endpoint of β̃ at infinity, this half-plane will be contained in H_{+} or H_{−}, contradicting the assumption that it contains P. It follows that if ā meets two regions P and P′ then each component of H^{2}\(H_{a}^{+}∪H_{a}^{−}) meets just one endpoint of β̃. Similarly, each P can only be connected directly to two other regions by train routes in K, for if there were more they would define three disjoint arcs at infinity, each of which contains one of the endpoints of β̃.

Figure 5. Part of the region K, outlined in solid lines. The dotted lines indicate a few of the halfplanes comprising H_{+} and H_{−}.

We conclude that K can be described as a chain of closed polygonal regions joined cusp-to-cusp by train routes (possibly with additional arcs meeting only a single region); see figure 5. Thus after adding diagonal arcs to each region, we can describe a path β̂ which passes through these diagonals and through the connecting train-routes. This train route in the extended track is again quasi-geodesic, and since the chain of regions was uniquely determined, invariant by the transformation T_{β} (clearly we may choose the added diagonals invariant). Hence it projects to a closed loop in S, carried on the projection of the extended track and obviously homotopic to β.

It remains just to check that the projected extension is still a train track, i.e. that none of the diagonals cross each other. But this is the same as checking that, if we translate the construction upstairs by π_{1}(S), no region is traversed by two crossing diagonals. If this were to happen we clearly would obtain two translates of β̃ whose endpoints separate each other, which would contradict the assumption that β is simple.

This concludes the proof in the case that τ′ = τ. Now in general, note that we have so far proved the following: If β is not carried in any extension of τ, then there is an edge e of τ whose lift ẽ has the property that all train routes of τ̃ through ẽ separate ∂β̃. Now consider a train route r′ of the lift τ̃′ of τ′ which passes through ẽ. Since τ̃′ is a diagonal extension of τ̃, r′ must be sandwiched in between two routes of τ̃ that pass through ẽ. It follows that r′ also separates ∂β̃. Hence when α is carried on τ′, it lifts to α(e) routes of τ̃′ through ẽ, and as before we obtain i(α, β) ≥ α(e).

3.3. Infinite diameter. We now have sufficient tools to prove the infinite-diameter claim in Theorem 1.1.

Proposition 3.6. diam(C_{1}(S)) = ∞.

Proof. Let h : S → S be a pseudo-Anosov homeomorphism, with stable lamination μ and unstable lamination ν. (For more about pseudo-Anosov homeomorphisms, see e.g. [9, 36, 2].) Thus h takes μ to λμ where λ > 1, and in fact [μ] is an attracting fixed point for h in the projectivized space PML(S) = (ML(S) \ {0})/R_{+}, such that every point in PML(S) \ [ν] approaches [μ] under iteration of h. Furthermore (see [9, Exposé 13] or [28]), h may be chosen so that the complementary regions of μ are either ideal triangles or punctured monogons.

Let τ_{0} be a train-track formed from a regular neighborhood of μ – then τ_{0} is necessarily maximal, that is its complementary components are triangles and punctured monogons, and are in one-to-one correspondence with those of μ.

One can homotope h to a standard form in which it permutes the complementary regions of μ, is expanding on the leaves of μ, and contracting in the transverse direction near μ. Thus, the image train-track h(τ_{0}) must be carried in τ_{0}. If the regular neighborhood is taken sufficiently small then P (τ_{0}) excludes ν, and therefore every cycle in τ_{0} approaches μ under iteration of h. In particular for large enough m and for every vertex cycle v of τ_{0}, h^{m}(v) runs through every branch of τ_{0}, and the weights that h^{m}(v) puts on each branch of τ_{0} go to infinity exponentially fast (in fact asymptotically as λ^{m}).

For large enough m we conclude that P (h^{m}(τ_{0})) ⊂ int(P (τ_{0})). Thus letting τ_{j} = h^{mj}(τ_{0}) we find by induction that P (τ_{j+1}) ⊂ int(P (τ_{j})). Hence if α_{j} ∈ P (τ_{j}) and β ∉ P (τ_{0}), Lemma 3.4 applied inductively shows that d_{C}(α_{j}, β) ≥ j (the maximality of τ_{0} implies that P E(τ_{j}) = P (τ_{j})). This concludes the proof.

3.4. The nesting lemma. We will need the following notation. If α ∈ C_{0}(S) and σ, τ are train tracks, let d_{C}(α, σ) denote min_{v} d_{C}(α, v) and d_{C}(σ, τ) = min_{v,w} d_{C}(v, w), where v ranges over the vertices of σ and w ranges over the vertices of τ.

The goal of this section is the following lemma, whose proof will appear at the end of it.

Lemma 3.7. (Nesting Lemma) There exists a D_{2} > 0 such that, whenever ω ≺ τ is a nested pair of large recurrent tracks and d_{C}(ω, τ) ≥ D_{2}, we have

P N(ω) ⊂ int(P N(τ)).

Given a train-track τ and a measure μ ∈ P (τ) we can define a combinatorial length ℓ_{τ}(μ) as ∑_{b} μ(b), where the sum is over the branches b of τ. Similarly if μ ∈ P N(τ) we can define ℓ_{N(τ)}(μ) as the minimum of combinatorial lengths in the tracks of N(τ) that carry μ.

There are some easy consequences of there being only finitely many combinatorial types of train-tracks on S. For example, if λ ∈ P (τ) and one writes λ as a combination ∑_{i} a_{i}α_{i} (not necessarily unique) of the vertices α_{i} of τ with nonnegative coefficients, then

max_{i} a_{i} ≤ ℓ_{τ}(λ) ≤ C_{1} max_{i} a_{i} (3.2)

where C_{1} depends on a bound on the number of vertices of τ, and a bound C_{0} for ℓ_{κ}(ω) over all train tracks κ and vertices ω.

Another consequence of finiteness is that there is a constant B depending only on S, such that any two vertices of a train-track are C-distance at most B apart. (We conjecture that B = 2).

Furthermore we have:

Lemma 3.8. Given L > 0 there exists D_{0}(L) so that, if α ∈ P (τ) and d_{C}(α, τ) ≥ D_{0}(L) then ℓ_{τ}(α) ≥ L.

Proof. Fixing L and τ, only finitely many curves α are carried by τ with ℓ_{τ}(α) ≤ L. Thus there is an upper bound on their distance from the vertices of τ. Taking a maximum over all combinatorial types of train-tracks in S, we have the desired statement.

We also observe:

Lemma 3.9. If α ∈ P (τ) and d_{C}(α, τ) ≥ 3 then α fills a large subtrack of τ.

Proof. Suppose that α is carried in κ < τ which is not large. Then S \ κ contains a nontrivial, nonperipheral curve β, so that d_{C}(β, α) ≤ 1 and d_{C}(β, v) ≤ 1 for any vertex of κ. By the triangle inequality d_{C}(α, v) ≤ 2, and since v is also a vertex of τ, d_{C}(α, τ) ≤ 2.

The next lemma addresses the following issue. A closed curve carried on an extension of a track σ does not necessarily trace through any complete cycle on σ. However, if σ is sufficiently deeply nested in τ, then any curve on an extension of σ is forced to run through a cycle of τ, and in fact must put a definite amount of weight on that cycle.

Lemma 3.10. There exists M_{0}, and for any L there exists D_{1}(L) such that if σ is large, σ ≺ τ, and d(σ, τ) ≥ D_{1}(L) then the following holds. Suppose σ′ ∈ E(σ) and τ′ ∈ E(τ), and σ′ ≺ τ′. Then any curve β carried on σ′ can be expressed in P (τ′) as β_{τ} + β′_{τ}, where β_{τ} ∈ P (τ), and

ℓ_{τ′}(β′_{τ}) ≤ M_{0}ℓ_{σ′}(β), (3.3)

ℓ_{τ}(β_{τ}) ≥ Lℓ_{σ′}(β). (3.4)

Proof. It suffices to prove the lemma when β is a vertex v of σ′. For the general case, express β as a combination of vertices and use (3.2).

Let W_{0} be a bound (by finiteness) on the weights that v puts on any branch of σ′, so that by Lemma 3.3 v puts at most m_{0}W_{0} on the branches of τ′ \ τ. Write the vertices of τ′ as {α_{i}} ∪ {γ_{j}}, where α_{i} are the ones supported in τ. Then in the coordinates of P (τ′) we may write v = v_{τ} + v′_{τ} where v_{τ} = ∑ a_{i}α_{i} and v′_{τ} = ∑_{j} c_{j}γ_{j}, with a_{i}, c_{j} ≥ 0.

For each branch b of τ′ \ τ we have v(b) = ∑ c_{j}γ_{j}(b) ≤ m_{0}W_{0}. Since for each j some b has γ_{j}(b) ≥ 1, we have c_{j} ≤ m_{0}W_{0}.

We have shown

ℓ_{τ′}(v′_{τ}) ≤ m_{0}W_{0}C_{0}.

Letting M_{0} = m_{0}W_{0}C_{0}, and noting that ℓ_{σ′}(v) ≥ 1, we have the first desired inequality (3.3). Now on the other hand, let D_{1} = 2B + D_{0} where D_{0} = D_{0}(C_{0}L + M_{0}) is given by lemma 3.8.

Since the distance between a vertex of σ′ and any vertex of σ is at most B, and the same for τ′ and τ, we conclude from the assumption d_{C}(σ, τ) ≥ D_{1} that we have d_{C}(v, τ′) ≥ D_{0}, and by Lemma 3.8, we have ℓ_{τ′}(v) ≥ C_{0}L + M_{0}. Now ℓ_{τ}(v_{τ}) = ℓ_{τ′}(v)−ℓ_{τ′}(v′_{τ}) ≥ C_{0}L, and since ℓ_{σ′}(v) ≤ C_{0} we have the second inequality (3.4).

Proof of Lemma 3.7 (Nesting Lemma). Let ω ≺ τ with d_{C}(ω, τ) ≥ D_{2}, where D_{2} will be determined shortly. Let σ be any large subtrack of ω. We will prove that P E(σ) ⊂ int(P E(κ)) for some large subtrack κ of τ. Thus by definition we will have P N(ω) ⊂ int(P N(τ)), which is the desired statement.

We may assume that σ fills τ. If not, replace τ by the smallest subtrack carrying σ, which must necessarily be large.

Let τ = τ_{0} and ⋯ τ_{2} ≺ τ_{1} ≺ τ_{0} be a sequence of tracks obtained from τ_{0} by splitting, so that σ ≺ τ_{j} for each j. Let ρ be the first τ_{j} for which d_{C}(τ_{j}, τ) > 2. By the properties of splitting sequences (see Section 3.1), τ_{j} either shares a vertex with τ_{j−1} or is a subtrack of a track that shares a vertex with it. Thus for any vertex v of ρ = τ_{j}, d_{C}(v, τ_{j+1}) ≤ B, and it follows that d_{C}(v, τ) ≤ 2 + 2B. Therefore d_{C}(σ, ρ) ≥ D_{2} − 2 − 2B.

Fix now β carried by σ′ ∈ E(σ), and let us show that β ∈ int(P E(κ)) for some large subtrack κ < τ. The idea will be that, by Lemma 3.10, β will place definite weight on some cycle of ρ, and by Lemma 3.9 this cycle will fill a large subtrack κ_{0} of τ. On the other hand β will place relatively little weight on any extension branches outside τ, and we will be able to reach our conclusion for some κ containing κ_{0}.

Since σ fills ρ, by Lemma 3.2 there is some ρ′ ∈ E(ρ) carrying β. Fix L_{1} (to be determined shortly). Lemma 3.10, together with (3.2), imply that for sufficiently high D_{2} we can write β = β_{ρ} + β′_{ρ} where β_{ρ} = ∑ a_{i}α_{i} over vertices α_{i} of ρ, and at least one of the a_{i} > L_{1}. Now applying Lemma 3.9, since d_{C}(α_{i}, τ) > 2 there is a large subtrack κ_{0} of τ such that α_{i}(b) ≥ 1 for each branch b of κ_{0}.

It follows that β(b) ≥ L_{1}ℓ_{σ′}(β) for all branches b of κ_{0}, but we don’t know if β ∈ P E(κ). The trouble is that the enlargement of κ_{0} that supports β may not be a diagonal extension. Thus we will find an intermediate track between τ and κ_{0} by adding branches to κ_{0} that have too much weight to be pushed to the corner, and show that this process terminates with the desired track.

Let τ′ ∈ E(τ) be a track carrying β (by Lemma 3.2). By Lemma 3.3 we know that β(b) ≤ m_{0}ℓ_{σ′}(β) for any branch b of τ′ \ τ.

If for all branches c of τ′ \ κ_{0} which meet κ_{0} we have β(c) < δL_{1}ℓ_{σ′}(β) then by Lemma 3.1, β ∈ int(P E(κ_{0})) and we are done. If not, let c violate this inequality, and define an extension κ_{1} of κ_{0} containing c as follows: let c_{±} be the ends of c where c_{−} ∈ κ_{0}. If c_{+} ∈ κ_{0} then κ_{1} = κ_{0} ∪ c is a train track. If not, then c_{+} is incoming to some switch with at most m_{1} branches outgoing (m_{1} = m_{1}(S)). At least one of those, c_{1}, has measure β(c_{1}) ≥ \frac{1}{m_{1}} β(c) ≥ \frac{δ}{m_{1}} L_{1}ℓ_{σ′}(β). Add this branch, and continue adding branches until we find one which touches κ_{0} again. Let κ_{1} denote κ_{0} together with this chain of branches, and note that for all branches b of κ_{1}, β(b) ≥ \frac{δ}{m_{2}} L_{1}ℓ_{σ′}(β), where m_{2} = m_{2}(S).

If now there is no edge of τ′\κ_{1} adjacent to κ_{1} with measure at least δ \frac{δ}{m_{2}} L_{1}ℓ_{σ′}(β), we are done. Otherwise, we can repeat this process, obtaining a sequence κ_{i} of extensions, which must terminate after at most m_{3} steps. Thus, β(b) for any branch b of κ_{i} is always at least (\frac{δ}{m_{2}} )^{m_{3}} L_{1}ℓ_{σ′}(β). If we have chosen L_{1} sufficiently

large that m_{0} < δ (\frac{δ}{m_{2}} )^{m_{3}} L_{1}, this process must terminate without appending to κ_{j} any branches of τ′ \ τ. Therefore we must end with some κ_{j} < τ for which β ∈ int(P E(κ_{j})), and we are done.

We note that a corollary of the proof is the following quantitative version of the Nesting Lemma, obtained by taking the constant L_{1} sufficiently large:

Lemma 3.11. The constant D_{2} in the Nesting Lemma may be chosen so that, if ω ≺ τ and d_{C}(ω, τ) ≥ D_{2}, then for any β ∈ P N(ω) there is a subtrack κ < τ such that β ∈ P E(κ) and, for any branch b of κ,

β(b) ≥ 2ℓ_{N(ω)}(β).

3.5. Growth of intersection numbers. Lemma 3.11 implies, in particular, that the combinatorial length of a curve carried on a train track grows exponentially with its distance from a fixed point, say a vertex of the track. As a consequence (see also Lemma 3.5), its intersection number with any fixed curve not carried on the track should grow exponentially.

A finer analysis shows that, if two curves are both far from a fixed one and relatively near each other, then both are deeply nested in diagonal extensions of the same track, and as a consequence their intersection numbers with any fixed curve are very large compared to their intersection number with each other. In the closing argument of the Projection Theorem 2.6, in Section 5, we will use the following quantitative version of this observation.

Lemma 3.12. Given Q, k > 0 there exist D_{3}, ν such that the following holds. If α, β and γ in C(S) are such that

d_{C}(β, α) ≥ D_{3}

and

d_{C}(γ, β) ≤ νd_{C}(β, α)

then

min_{α′} i(β, α′) · min_{α′} i(γ, α′) ≥ Qi(β, γ),

where α′ varies over the k-neighborhood of α in C.

Proof. Extend α to a pair of pants decomposition of S. Such a decomposition determines a family of standard train-tracks, obtained by choosing one of a finite number of configurations in each pair of pants and in a connecting collar between any adjacent pairs of pants. (See Penner-Harer [37]). This family of tracks has the property that any simple closed curve on S is carried by one of them. Thus, let τ_{0} denote a standard train-track carrying β. Depending on the choice of local picture in an annulus neighborhood of α, α is either a vertex cycle of τ_{0}, or is distance at most 2 from a vertex cycle. It follows that d_{C}(β, τ_{0}) ≥ d_{C}(β, α) − (2 + B).

Since β is assumed far from α, by Lemma 3.9 it fills a large subtrack of τ_{0}, which we will continue to call τ_{0}.

Now we can find a sequence τ_{n} ≺ ⋯ ≺ τ_{0} = τ of train-tracks, each carrying β, so that d_{C}(τ_{j+1}, τ_{j}) > D_{2} and the length of the sequence is n ≥ d_{C}(β, τ)/(D_{2} + 2B). This is done by splitting. Perform a sequence of splittings of τ_{0} determined by the weights of β, and terminating with a track that has β as a vertex. Define inductively τ_{j+1} to be the first track in the sequence such that d_{C}(τ_{j+1}, τ_{j}) > D_{2}. Then, as in the proof of Lemma 3.7, d_{C}(β, τ_{j+1}) ≥ d(β, τ_{j}) − (D_{2} + 2B), and we may continue.

Lemma 3.7 now guarantees that P N(τ_{j+1}) ⊂ int(P N(τ_{j})).

If d_{C}(α, α′) ≤ k then d_{C}(α′, τ_{0}) ≤ k + 2, and by applying Lemma 3.4 inductively we see that α′ cannot be in P N(τ_{k+3}).

Another application of Lemma 3.4 shows that if d_{C}(β, γ) ≤ m < n, then γ ∈ P N(τ_{n−m}).

Thus, assuming n is sufficiently large compared to m, we may conclude that both β and γ are contained in P N(τ_{k+3}). Furthermore, applying Lemma 3.11 repeatedly, we also have that for a large subtrack κ of τ_{k+3}, γ puts weight at least 2^{n−m−k−3}ℓ_{N(τ_{n−m})}(γ) on every branch of κ. The same holds for β and a large subtrack κ′ of τ_{k+3}. Applying Lemma 3.5, we find that

i(α′, γ) ≥ 2^{n−m−k−3}ℓ_{N(τ_{n−m})}(γ),

and similarly for β.

On the other hand it is easy to see that

i(β, γ) ≤ C_{2}ℓ_{N(τ_{n−m})}(β)ℓ_{N(τ_{n−m})}(γ).

where C_{2} depends only on the topological type of S. This is because in every branch of τ_{n−m} a strand of β and one of γ can only have one essential intersection, and a strand in a diagonal branch of N(τ_{n−m}) can only hit strands in diagonal branches, and two diagonal branches can intersect at most twice.

Putting these inequalities together, if n − m − k is sufficiently high we have the desired inequality.

4. Geometry of quadratic differentials

Let q be a holomorphic quadratic differential of area 1 with respect to some conformal structure x on S. In this section we will study the geometry imposed by q, with particular regard to the way nearly horizontal and nearly vertical geodesics are arranged, and how they intersect each other. Our main goals are the Vertical Domain Lemma 4.6, which gives a particular “thickening” of a nearly vertical curve with some useful properties, and the Intersection Number Lemma 4.8, which gives conditions for a nearly vertical and a nearly horizontal curve to have large intersection number.

4.1. Basic properties and uniform estimates. A straight segment with respect to q is a path containing no singularities in its interior, and which is geodesic in the locally Euclidean metric of q. A general geodesic segment is composed of straight segments which meet at singularities making an angle of at least π on either side. A straight segment connecting two singularities is also called a saddle connection.

A metric cylinder in q is an annulus which is isometric to the product of a circle and a line segment.

When S has no punctures, each nontrivial homotopy class has a geodesic representative. However when there are punctures the metric of q is incomplete and we must slightly generalize the notion. From now on by “geodesic representative” of a closed curve α we mean a curve α^{∗} in the compactified surface Ŝ (adding the punctures) such that α^{∗} ∩ S is composed of geodesic arcs, and there is a homotopy from α to α^{∗} which until the last moment is contained in S. It is not hard to see that any non-peripheral homotopy class has such a geodesic representative, which has minimal length, and the representative is unique unless there is a metric cylinder foliated by curves in the homotopy class.

Topological constants. For later reference, n_{1}, …, n_{5} will denote the following bounds, which may easily be computed in terms of the genus and number of punctures of S. Let n_{1} bound the number of singularities of q, including punctures. Let n_{2} bound the number of disjoint saddle connections which may appear simultaneously in S. Let n_{3} be an isoperimetric constant, such that Area_{q}(X) ≤ n_{3} diam_{q}(X)^{2} for any subset X of S. Let n_{4} bound the size of a sequence X_{1} ⊂ ⋯ ⊂ X_{n_{4}} ⊂ S for which i_{∗}π_{1}(X_{j}) is a proper subgroup of i_{∗}π_{1}(X_{j+1}), where i_{∗} is the map induced on π_{1} by inclusion into S. Let n_{5} bound 1/π times the sum of cone angles over all singularities of q.

Definite collars. Let the width of an annulus A in a metric q denote the minimal distance between boundaries, and the circumference the minimal length of a curve going once around A. A compactness argument using the moduli space of Riemann surfaces yields the following:

Lemma 4.1. There exists W > 0 depending only on the topology of S such that, for each unit area quadratic differential q there exists a nonperipheral annulus of width W. Furthermore, given μ there exists L > 0 so that the annulus can be chosen either to be a metric cylinder of modulus at least μ, or to have circumference at least L.

Proof. If the statement is false, then there is a sequence of conformal structures x_{i} on S, unit-area holomorphic quadratic differentials q_{i}, and L_{i} → 0, W_{i} → 0 such that there is no annulus in (S, q_{i}) of width at least W_{i} which either has circumference at least L_{i} or is a metric cylinder of modulus μ.

We can now apply a compactification argument whose details may be found in Masur [27]. We may take a subsequence so that (S, x_{i}) converge in a compactified moduli space to a noded Riemann surface (S′, x), where S′ may be taken as the complement in S of a collection of disjoint curves, and q_{i} converge on compact sets of S′ to some q. Given μ > 0 there is a K(μ) > 0 (depending on the topological type of S) such that the following alternative holds: if diam(q_{i}) ≥ K for all sufficiently high i then eventually (S, x_{i}, q_{i}) contains a metric cylinder of width at least W, and modulus at least μ. In this case we have contradicted the choice of sequence, hence we are done. If diam(q) ≤ K for all sufficiently high i then the limiting q is non-zero on at least one component R of S′. (The two possibilities are not mutually exclusive). We also note that q has at most simple pole singularities at the punctures.

Since R supports a non-zero holomorphic quadratic differential of finite area, it cannot be a sphere with less than 4 punctures. It follows that there is some simple, nontrivial, nonperipheral curve in R, so let A be any collar for this curve. If W and L are the width and circumference of A, then in the approximating metrics of q_{i} for high enough i we obtain annuli of nearly these width and circumferences, again contradicting the choice of sequence.

Definite boxes. We will need the following notion, where a rectangle denotes an embedded Euclidean rectangle with respect to q, in particular containing no singularities in its interior.

Definition 4.2. Let ω denote a q-geodesic segment or closed curve. If N, δ > 0, an (N, δ) box for ω is a rectangle containing at least N parallel strands of ω (counting multiplicity) of equal length δ, parallel to two of the sides of the rectangle. The endpoints of the strands are on the orthogonal sides of the rectangle. The lengths of the orthogonal sides are at most δ.

Remark: Note that if N < 1, an (N, δ) box means a (1, δ)-box.

As a consequence of Lemma 4.1 we can prove the following:

Lemma 4.3. Let q be a unit-area holomorphic quadratic differential on (S, x), and suppose that there are no q-metric cylinders of modulus greater than 2 in S. Let A denote the nonperipheral annulus of width W and length L provided by Lemma 4.1. There exist δ, r > 0, depending only on the topology of S, such that for any closed q-geodesic γ which has intersection number N > 0 with the core of A, there is a (rN, δ)-box for γ in A. Furthermore, the q-injectivity radius at the center of the box is at least δ.

Proof. On a smaller annulus A′ ⊂ A of width W/2 the q-injectivity radius is at least δ_{1} = min(W/4, L/2). There are N segments (with multiplicity) of γ of length W/2 passing through A′. Centered on any nonsingular point of a segment σ of γ ∩ A′ there is a geodesic segment orthogonal to σ of length 2δ_{1}, and so (recalling n_{1} from above) there must be a segment on σ of length at least W/2n_{1} for which these orthogonal segments meet no singularities, and therefore make a (1, W/2n_{1}) box for σ, with center on σ

Consider all such boxes in A. There are N (with multiplicity) and we must check that there is sufficient overlap. For each box consider a box of half the size with the same center. Since each box has definite area and the area of q is 1, we find that there must be a point simultaneously in max(rN, 1) half-boxes for a fixed r > 0, and hence a box containing max(rN, 1) centers of boxes. This is the desired box.

4.2. Vertical and horizontal. From now on, let us suppose that two constants θ, ϵ > 0 have been fixed satisfying a short list of constraints which will appear in the course of the proof. For now assume θ < min(1/2, ϵ^{2}).

Definition 4.4. We say a straight segment is almost vertical (respectively almost horizontal) with respect to q if its direction is within θ of the vertical (resp. horizontal) direction of q. We say a geodesic segment or closed curve is almost vertical (resp. almost horizontal) if it is composed of straight segments each of which is almost vertical (resp. almost horizontal) or has length at most ϵ.

Note that a (weak) consequence of the condition θ < 1/2 is that

|α|_{q,v} > \frac{1}{2}|α|_{q}.

We now define a certain type of thickening, which we call a vertical (or horizontal) domain, that will be useful in several places.

Definition 4.5. Let ω be an almost vertical geodesic segment or closed curve. The vertical domain Ω_{ϵ}(ω) is constructed as follows. For any point p ∈ ω let σ_{p} be the maximal open q-horizontal segment which contains no singularities or punctures, and such that each component of σ_{p} − {p} has length at most ϵ. Let Ω_{ϵ}(ω) be the closure of ∪_{p∈ω}σ_{p}.

We similarly define a horizontal domain Ψ_{ϵ}(ω) if ω is almost horizontal, where the σ_{p} are vertical segments.

Let us record some useful properties of this construction.

Figure 6. An example of a vertical domain. The horizontal foliation is dotted, ω is solid and Ω_{ϵ}(ω) is in grey

Lemma 4.6. There exist positive L_{0} and a_{0}, a_{1}, a_{2} such that the following holds. Let q be a unit-area quadratic differential on S and let ω be an almost vertical geodesic segment or closed curve. If ω is a segment, assume it has length at least L_{0}. Let τ be an almost-horizontal segment of diameter diam_{q}(τ) ≥ a_{0}ϵ, which is disjoint from ω. Let Ω = Ω_{ϵ}(ω). Then we have:

(1) The map π_{1}(Ω) → π_{1}(S) induced by inclusion has non-trivial, non-peripheral image.

(2) There is a subsegment of τ of diameter at least a_{1} diam_{q}(τ) which is disjoint from Ω.

(3) The boundary of Ω has q-length at most a_{2}ϵ + 1/ϵ

(4) The boundary of Ω has horizontal length at most a_{2}ϵ.

Proof. Let n_{1}, n_{2} and n_{5} be the topological constants described in §4.1, and choose L_{0} = n_{2}ϵ + 3/ϵ. We will see that in fact the a_{i} can be written explicitly in terms of the n_{i}.

Part (1) is obvious if ω is a closed geodesic, so suppose otherwise. For all nonsingular p ∈ ω let σ_{p} be the horizontal arcs of Definition 4.5. For all y ∈ Ω let f(y) = #{p : y ∈ σ_{p}}. Then Ω can be described as a union of closed parallelograms of width 2ϵ, whose heights sum to |ω|_{q,v}, and f gives the degree of overlap of interiors of these parallelograms. It follows that ∫_{Ω} f(y) = 2ϵ|ω|_{q,v} (where the integral is with respect to q-area) On the other hand ∫_{Ω} f(y) ≤ max(f) Area(Ω). Since the area of q is 1 we conclude

max f ≥ 2ϵ|ω|_{q,v}.

If ω′ is the union of almost vertical straight segments of ω and ω′′ is the rest, then |ω′′|_{q} ≤ n_{2}ϵ, so |ω′|_{q} ≥ L_{0} − n_{2}ϵ. Since |ω′|_{q,v} ≥ \frac{1}{2}|ω′|_{q}, the choice of L_{0} guarantees that max f ≥ 3.

We conclude that there is a point y contained in a horizontal segment σ ⊂ Ω which cuts ω in three places. If there are two consecutive intersection points on σ with the same orientation, then a segment of ω together with an interval of σ make a simple curve in Ω, geodesic except at the intersection points where the total turning angle (measured in the q metric) is at most 2θ < π, and hence it cannot be trivial or peripheral. (A curve bounding a disk has total turning angle at least 2π, and a curve bounding a puncture has total angle at least π.) If every two consecutive intersections have opposite orientations, we can take three consecutive points so that the orientation matches on the outer two, produce a curve passing through a segment of σ and ω with one self-intersection point, and do surgery to get a simple curve with total turning angle at most 4θ < π. Again it must be non-trivial and non-peripheral. This proves part (1).

We now consider parts (3) and (4). For any y ∈ ∂Ω, there is some z ∈ ω joined to y by a horizontal arc σ which meets ω only in z. Note that σ may pass through a singularity or a puncture (possibly z itself). If so, then a portion of σ may lie on ∂Ω, and contributes at most ϵ to |∂Ω|_{q,h}. The number of horizontal arcs issuing from a singularity is 1/π times its cone angle, so this contribution to |∂Ω|_{q,h} is bounded by n_{5}ϵ.

If σ meets no singularities then y is contained in a segment of ∂Ω parallel to a segment of ω containing z. The total length of such portions of ∂Ω which are not almost vertical is therefore bounded by 2n_{2}ϵ: there are at most n_{2} such segments in ω, they can be approached from either side, and each has length at most ϵ.

Finally for any segment κ of the portion of ∂Ω which is almost vertical, we note that the segments σ form an embedded parallelogram of width ϵ and height |κ|_{q,v}. Since q is unit area, we conclude that the vertical length of this portion of the boundary is at most 1/ϵ. Its horizontal length is bounded by (1/ϵ) tan θ < 2θ/ϵ (assuming θ < 1/2), which is bounded by 2ϵ since θ < ϵ^{2}. Putting these together we have a bound |∂Ω|_{q,h} ≤ (n_{5} + 2n_{2} + 2)ϵ, and |∂Ω|_{q} ≤ (n_{5} + 2n_{2} + 2)ϵ + 1/ϵ, which proves parts (3) and (4).

Finally it remains to prove (2). For y ∈ τ ∩ Ω, let σ_{y} be the horizontal segment of length at most ϵ joining y to ω. Suppose that y is at least 2ϵ away from any singularity of q, endpoint of ω or τ, or segment of ω that is not almost vertical. Then (with the assumption θ < 1/2) a segment of the almost-horizontal τ of length 2ϵ must intersect the almost-vertical segment of ω passing through the endpoint of σ, but we have assumed τ ∩ω = ∅. Thus, τ ∩Ω is contained in a 2ϵ-neighborhood of the singularities, endpoints and non-almost-vertical segments of ω. There are at most k = n_{1} + n_{2} + 2 of these, each of diameter at most ϵ. Let d be the largest diameter of a component of τ \ Ω. Then the diameter of τ is bounded by 5ϵk + (k + 1)d (the 5ϵ bounds the diameter of a 2ϵ neighborhood of a segment of diameter ϵ). For diam_{q}(τ) ≥ 10ϵk, say, we find that d ≥ diam_{q}(τ)/2(k + 1), which gives part (2).

Let us also note the following observation which will be used in the proofs of Lemmas 4.8 and 5.5.

Lemma 4.7. If τ is an embedded segment in a disk D in S and q is a holomorphic quadratic differential, then |∂D|_{q,h} ≥ 2|τ|_{q,h}. If τ is in a once-punctured disk D′ and q has at most a simple pole at the puncture then |∂D′|_{q,h} ≥ |τ|_{q,h}.

Proof. Consider first a disk D. For any p ∈ τ extend a vertical segment σ in both directions until it hits ∂D. This will happen since D is simply connected. It follows immediately that the horizontal length of the portion of ∂D cut off by these segments is equal to twice the horizontal length of D. For a punctured disk D′, note that a segment σ could hit τ at both ends, on the same side of τ, if it goes around the puncture. However since the puncture is at most a pole this can only happen on one side of τ, and the vertical segments extended from the other side still give the desired bound.

4.3. The intersection number lemma. Our first application of the vertical domain and box lemmas is the following lemma, which states that an almost horizontal and an almost vertical curve which intersect every bounded curve a definite amount also intersect each other proportionally. Compare this fact with Lemma 3.12; the two will be applied together to yield a contradiction.

Lemma 4.8. Suppose that ϵ < min(δ/4, a_{1}^{n_{4}}δ/2a_{0}, a_{1}^{n_{4}}δ/2n_{4}a_{2}), in addition to previous constraints. There exist M, h > 0 such that, given x ∈ T (S) and unit area quadratic differential q on (S, x), if β is an almost horizontal closed geodesic and γ is an almost vertical closed geodesic with respect to q, and for every non-peripheral simple closed α of q-length at most M,

i(β, α) ≥ B and i(γ, α) ≥ C, (4.1)

then

i(β, γ) ≥ hBC

Proof. Apply lemma 4.1 to get constants L, W > 0 such that either q has a flat cylinder of modulus 2, or an annulus of circumference at least L and radius W. Let M = max(1/\sqrt{2}, 1/W, n_{4}(a_{2}ϵ + 1/ϵ)). Consider now both possible cases. Case A. If q has a flat cylinder of modulus 2, let α be the core of this cylinder. Then α has q-length at most 1/\sqrt{2}, so that (4.1) gives B and C strands of β and γ, respectively, crossing the annulus. It is easy to see that any two nearly orthogonal segments cutting through the annulus must intersect at least once. It follows that i(β, γ) ≥ BC, so with h ≤ 1, we are done. Case B. If q has an annulus A with circumference at least L and width W, note that A has modulus at least W^{2} (since its area is at most 1), and hence Ext_{x}(α) ≤ 1/W^{2}, where α is the core of A. In particular |α^{∗}|_{q} ≤ 1/W where α^{∗} is the q-geodesic representative (see §2.3). Thus γ contains at least C segments (with multiplicity) crossing A and hence of length at least W, and similarly β contains at least B such segments.

Lemma 4.3 guarantees an almost horizontal (cB, δ)-box H for β, with injectivity radius at least δ at its center. Let τ_{0} denote the segment of length δ/2 centered on the center of H, parallel to the direction of β. It has the property (recalling θ < 1/2) that any almost-vertical segment that meets τ_{0} must cut through all the β strands in H.

Since γ must have length at least CW, we may divide it into at least CW/L_{0} − 1 pieces, each of which has length at least L_{0}. (If CW/L_{0} < 2 we may instead take the whole closed curve γ, and prove the theorem for C = 1. The discrepancy is absorbed in the constants.) Each of these is almost vertical, though they may traverse saddle connections of length at most ϵ which are not almost vertical. However, these short segments cannot meet τ_{0}, since H contains no singularities (here we are using the assumption ϵ < δ/4). Thus, for each segment that meets τ_{0} we obtain rB essential intersections with β. If all of them do meet τ_{0}, then we are done.

Thus suppose that one segment ω_{1} is disjoint from τ_{0}. Let X_{1} = Ω_{ϵ}(ω_{1}). Lemma 4.6 guarantees that X_{1} generates a nontrivial, nonperipheral subgroup of π_{1}(S). Note that the diameter of τ_{0} is δ/2, by the injectivity radius lower bound in H. Thus part (2) of Lemma 4.6 guarantees that a subarc τ_{1} of τ_{0}, with diameter a_{1}δ/2, is disjoint from X_{1} (to apply the Lemma we need δ/2 ≥ a_{0}ϵ, which is implied by the conditions on ϵ).

Because the q-length of any nontrivial nonperipheral component of ∂X_{1} is bounded by a_{2}ϵ + 1/ϵ ≤ M, γ intersects it essentially, C times. Thus for any component Y of S\X_{1} which is not a disk or punctured disk, there are C arcs of γ (with multiplicity) passing through Y with both endpoints on ∂Y, which are not deformable back into X_{1}.

Apply this where Y is the component of S \ X_{1} containing τ_{1}. This cannot be a disk or punctured disk, because the horizontal length of ∂Y, which is at most a_{2}ϵ by Lemma 4.6, is smaller than the length a_{1}δ/2 of τ_{1} by our assumptions on ϵ, and we may apply Lemma 4.7.

Thus, if all C arcs in Y meet τ_{1} then as before we have our required intersections between β and γ, and we are done.

If one arc ω_{2} is disjoint from τ_{1} then define X_{2} = X_{1} ∪ Ω_{ϵ}(ω_{2}). We may apply Lemma 4.6 and the same arguments as before to find a subarc τ_{2} of τ_{1}, of diameter a_{1}^{2}δ/2, which is disjoint from Ω_{ϵ}(ω_{2}), and hence from X_{2}.

We may continue by induction, generating a sequence X_{1} ⊂ ⋯ X_{j} ⊂ X_{j+1} and subarcs τ_{j} ⊂ τ_{1} of length a_{1}^{j}δ/2 disjoint from X_{j}. At each step, |∂X_{j}|_{q} is incremented by at most a_{2}ϵ + 1/ϵ, and |∂X_{j}|_{q,h} goes up by at most a_{2}ϵ. Since X_{j+1} cannot be deformed into X_{j} the process must terminate within n_{4} steps. By our assumption on ϵ we can apply Lemma 4.7 each time so that the component of S −X_{j} containing τ_{j} is never a disk or punctured disk. It follows that the only way the process can terminate is by giving C intersections of γ with τ_{j} for some j ≤ n_{4}, which concludes the proof.

5. Proof of the projection theorem

In this section we fix a holomorphic quadratic differential q of area 1 with respect to a conformal structure x_{0} on S, and let L_{q} denote the corresponding Teichmüller geodesic. We will denote the Riemann surfaces along L_{q} by x_{t} = L_{q}(t) where t is arclength, and the quadratic differentials by q_{t}.

Recall that the geodesic gives rise to a map F_{q} : R → C, and a projection π = π_{q} : C → R defined as in section 2. To prove the Projection Theorem 2.6 we must show that this projection satisfies the contraction property (Definition 2.2).

We will also assume that our constants ϵ, θ satisfy the assumptions of the Intersection Number Lemma 4.8.

5.1. Bounded adjustments. We will first need to examine transitions along L_{q} from mostly vertical to balanced to mostly horizontal curves. As measured by the Teichmüller length parameter, a nearly vertical curve can take a very long time to become balanced. However we find that in a number of crucial situations the transition takes bounded time as viewed in the curve complex (that is, when considering quantities such as diam_{C}(F [s, t]) instead of |s − t|).

The relevant insight is illustrated by this sketch of the proof of Lemma 5.5: Consider a very long nearly vertical segment with respect to q_{0}, which does not fill the whole surface (say it avoids a definite-length horizontal segment). Then if for t > 0 the segment is still long and nearly vertical, it fills up some proper subsurface of S which can only shrink as t increases. The boundaries of the resulting sequence of surfaces form a bounded-length sequence in C(S). This is made precise using the Vertical Domain construction.

Our first observation about the map F is that it is, on a large scale, Lipschitz:

Lemma 5.1. (Lipschitz) There exist C, D > 0 such that for any q, t_{1} and t_{2} we have

d_{C}(F_{q}(t_{1}), F_{q}(t_{2})) ≤ C|t_{2} − t_{1}| + D.

Proof. As in Lemma 2.4, let e_{0}(S) be such that for any conformal structure on S there is a curve with extremal length at most e_{0}. Suppose that |t_{2} − t_{1}| ≤ 1. Let α_{i} be a curve of shortest extremal length for x_{t_{i}}, for i = 1, 2. Then Ext_{x_{t_{1}}} (α_{2}) ≤ e^{2}e_{0} (by (2.1)). A bound on d_{C}(α_{1}, α_{2}) follows from Lemma 2.5.

The case where |t_{2} − t_{1}| > 1 follows by subdividing.

In the exceptional cases of projecting curves that are entirely horizontal or vertical, we observe the following:

Proposition 5.2. If β ∈ C \ C_{b}(q) then d_{C}(β, F_{q}(π_{q}(β))) ≤ 1.

Proof. Assume that β is vertical. Let Σ denote the union of compact singular leaves of the vertical foliation of q, and let Σ_{ϵ} denote a regular neighborhood of Σ. Then it is not hard to see (e.g. [32]) that for any non-peripheral curve γ in S the extremal length Ext_{x_{t}}(γ) remains bounded as t → +∞ if and only if γ can be deformed into Σ, and Ext_{x_{t}}(γ) → 0 as t → +∞ if and only if γ is homotopic to a boundary component of Σ_{ϵ}.

It follows that F_{q}(+∞) = F_{q}(π_{q}(β)) is one of these boundary components, and since β is in Σ, we obtain d_{C}(β, F_{q}(π_{q}(β))) ≤ 1. If β is horizontal we make a similar argument, reversing the t-direction.

The next lemma shows that the image under the map F of the set of t where a curve α is close to its minima has bounded diameter in C(S). In particular, once the q_{t}-length of α is sufficiently short, we only need to wait a bounded amount until it starts to grow again.

Lemma 5.3. There exist ϵ_{1}, d_{1} > 0, depending only on the topology of S, with the following property. If α is a closed q-geodesic homotopic to a simple curve, let

J = {t : |α|_{q_{t}} ≤ ϵ_{1}}.

Then diam_{C}(F (J)) ≤ d_{1}.

(Note that J is a bounded interval in R unless α is completely vertical or horizontal.)

Proof. Let ϵ_{1} = 2W. By Lemma 4.1, for each t there is a nonperipheral curve β_{t} with a collar neighborhood of q_{t}-width W. Since for t ∈ J, |α|_{q_{t}} ≤ 2W, we may conclude that i(α, β_{t}) = 0. Hence for any t, s ∈ J, d_{C}(β_{t}, β_{s}) ≤ 2.

The existence of the collar implies Ext_{x_{t}}(β_{t}) ≤ 1/W^{2}. Hence by Lemma 2.5, we conclude d_{C}(β_{t}, F (t)) ≤ 1/W^{2} + 1 for t ∈ J.

It follows that d_{C}(F (s), f(t)) ≤ 2/W^{2} + 4, and we set d_{1} accordingly.

The following lemma will allow us to convert a long almost-horizontal arc which has small diameter (i.e. winds around tightly) to one which has a definite diameter, within bounded distance in C.

Lemma 5.4. There exist constants ϵ_{3} > ϵ_{2} > 0 and d_{2} > 0, depending on the topology of S and the initial choice of ϵ, θ, so that the following holds. Let τ be an almost horizontal segment with respect to q, of length |τ|_{q} ≥ ϵ_{3}. Let J be the interval

J = {t ≥ 0 : diam_{q_{t}}(τ) < ϵ_{2}}

and suppose 0 ∈ J. Then diam_{C}(F (J)) ≤ d_{2}.

Proof. For any t let β_{t} be the homotopy class of the core of the annulus of width W given by Lemma 4.1. Let ϵ_{2} = W − 2ϵ.

Let Ψ^{t} denote the horizontal domain Ψ_{ϵ}(τ) with respect to q_{t} (see Definition 4.5). Then diam_{q_{t}}(Ψ^{t}) ≤ diam_{q_{t}}(τ) + 2ϵ < W for t ∈ J. Thus any closed curve in Ψ^{t} has 0 intersection number with β_{t}. We will show that if |τ|_{q} is sufficiently large, there exists a nontrivial, nonperipheral curve κ which is contained in Ψ^{t} for all t ∈ J.

We use an argument similar to the proof of Lemma 4.6 part (1). Recall the vertical intervals σ_{x} of radius ϵ around nonsingular x ∈ τ from the definition of Ψ^{0}. For any y ∈ S let f(y) = #{x ∈ τ : y ∈ σ_{x}}. Then ∫_{Ψ^{0}} f(y) with respect to q-area is wϵ|τ|_{q,h}. On the other hand the integral is at most Area_{q}(Ψ^{0}) max f, so that

max f ≥ \frac{2ϵ}{Area_{q}(Ψ^{0})}|τ|_{q,h} ≥ \frac{ϵ}{Area_{q}(Ψ^{0})}|τ|_{q},

where the second inequality is due to τ being almost horizontal. We also have Area_{q}(Ψ^{0}) ≤ n_{3} diam_{q}(Ψ^{0})^{2} ≤ n_{3}W^{2} where n_{3} was defined in §4.1. Thus we have max f ≥ ϵ|τ|_{q}/(n_{3}W^{2}). Set ϵ_{3} = 3n_{3}W^{2}/ϵ, and now |τ|_{q} ≥ ϵ_{3} implies max f ≥ 3. As in Lemma 4.6 we conclude that Ψ^{0} contains a nontrivial nonperipheral curve κ.

Since lengths in the vertical direction shrink as t increases, for all t ≥ 0 we have Ψ^{0} ⊂ Ψ^{t}. Thus κ is in all the Ψ^{t}, and hence must have 0 intersection number (hence C-distance 1) with all β_{t} for t ∈ J, as above. As in Lemma 5.3, d_{C}(β_{t}, F (t)) ≤ 1/W^{2} + 1, so it follows that F (t) is within bounded C-distance of κ for all t ∈ J.

The next lemma shows that if an almost vertical straight segment misses an almost horizontal segment of definite length, then after a bounded wait as measured in the curve complex, it will either be very short, or almost horizontal.

Lemma 5.5. Suppose x and q are given, and in addition to our previous assumptions we also have ϵ < min(ϵ_{2}/a_{0}, ϵ_{2}a_{1}/a_{2}). If α is a straight segment disjoint from an almost horizontal straight segment τ of length ϵ, let

J = {t ≥ 0 : |α|_{q_{t}} > ϵ_{1} and α not almost horizontal with respect to q_{t}}.

There is a number d_{3} = d_{3}(ϵ, θ) such that diam_{C}(F (J)) ≤ d_{3}.

Remark. Since ϵ_{2} was given as W − 2ϵ in Lemma 5.4, it is evident that our added conditions of the form ϵ < Cϵ_{2} are satisfied for ϵ sufficiently small.

Proof. If α is not almost vertical, then for a bounded T (depending on ϵ, θ) it will be almost horizontal with respect to q_{T}. By Lemma 5.1, F ([0, T]) has bounded diameter in C.

Thus we may assume α is almost vertical to begin. Suppose its length is at most L_{0}. Then for T = log 2L_{0}/ϵ_{1}, its q_{T} -vertical length is at most ϵ_{1}/2; thus it either has q_{T} -length less than ϵ_{1} or it is not almost vertical. In the first case we have satisfied the conclusion of the Lemma, again bounding diam_{C}(F ([0, T])) by Lemma 5.1. In the second case we are also done by the argument in the first paragraph.

Thus finally assume α is almost vertical and has length greater than L_{0}. Since |τ|_{q} = ϵ and τ is almost horizontal, for t_{1} = log 2ϵ_{3}/ϵ we have |τ|_{q_{t_{1}}} ≥ ϵ_{3}. Lemma 5.4 implies that either diam_{C}(F ([t_{1}, ∞))) ≤ d_{2} in which case we are done, or there is a t_{2} > t_{1}, with diam_{C}(F ([t_{1}, t_{2}])) ≤ d_{2}, so that diam_{q_{t_{2}}} (τ) ≥ ϵ_{2}.

Now if α is not almost vertical or has length at most L_{0} with respect to q_{t_{2}}, we are done by the above cases. Otherwise, we construct the vertical domain Ω_{1} = Ω_{ϵ}(α) with respect to q_{t_{2}}. Since α is disjoint from τ and (by assumption) ϵ_{2} > a_{0}ϵ, Lemma 4.6 gives a subarc τ_{1} of τ of length a_{1}ϵ_{2}, disjoint from Ω_{1}. The total horizontal length of ∂Ω is bounded by a_{2}ϵ by part (4) of Lemma 4.6. Thus, since a_{2}ϵ < a_{1}ϵ_{2} and applying Lemma 4.7, we conclude that the component Y of S \ Ω containing τ_{1} cannot be a disk or once-punctured disk. Thus ∂Ω_{1} has nontrivial and nonperipheral components. For each such component σ, we have |σ^{∗}|_{q_{t_{2}}} ≤ ℓ_{0} = a_{2}ϵ + 1/ϵ by part (3) of Lemma 4.6, where σ^{∗} is the geodesic representative. This bound means that within bounded Teichmüller distance either |σ^{∗}|_{q_{t}} reaches ϵ_{1}, or it starts to increase. Applying Lemma 5.3, we conclude that either the remaining diam_{C}(F ([t_{2}, ∞))) is bounded, in which case we are done, or there is t_{3} with bounded diam_{C}(F ([t_{2}, t_{3}])), such that for t > t_{3}, |σ^{∗}|_{q_{t}} > ϵ_{1}. It follows that for an additional t_{4} with t_{4} − t_{3} bounded, |σ^{∗}|_{q_{t_{4}}} ≥ 2ℓ_{0}.

We can now repeat the argument: There is a t_{5} such that diam_{C}(F ([t_{4}, t_{5}])) is bounded, so that with respect to q_{t_{5}} we either have the desired condition for α, or α is almost vertical, of length at least L_{0}, and τ_{1} now has diameter at least ϵ_{2}. Thus Lemma 4.6 again gives a vertical domain Ω_{2} = Ω_{ϵ}(α) with respect to q_{t_{5}} whose boundary components have q_{t_{5}}-length at most ℓ_{0}. Since our previous boundary components σ now have |σ^{∗}|_{q_{t_{5}}} ≥ 2ℓ_{0}, no nontrivial nonperipheral component of ∂Ω_{2} is homotopic to σ. The vertical domains decrease monotonically as t increases, so we conclude that i_{∗}(π_{1}(Ω_{2})) is a proper subgroup of i_{∗}(π_{1}(Ω_{1})).

We may repeat this procedure, obtaining a sequence Ω_{j+1} ⊂ Ω_{j} which terminates in at most n_{4} steps, at which point α has length less than ϵ_{1} or is almost horizontal, as desired, or we find that a remaining interval [t, ∞) has bounded-diameter image.

From now on let us assume that ϵ, θ satisfy the conditions of Lemma 5.5 as well as the previous conditions.

The following lemma shows that if a curve is balanced at L_{q}(0), then in the forward direction it will become almost horizontal after an interval of bounded size in the curve complex.

Lemma 5.6. (Almost Horizontal) Suppose β is balanced with respect to q. Let

J = {t ≥ 0 : β is not almost horizontal with respect to q_{t}}.

There exists d_{4} = d_{4}(ϵ, θ) such that diam_{C}(F (J)) ≤ d_{4}.

Proof. Let n_{3} be the bound for the number of mutually disjoint saddle connections one can have in S. Thus, β runs through at most n_{3} saddle connections, although some may be traversed arbitrarily many times.

Since |β|_{q,h} = |β|_{q,v}, for t > t_{1} = \frac{1}{2} log 1/θ we have |β|_{q_{t},h} > \frac{1}{θ}|β|_{q_{t},v}. It follows that if β_{0,t} is the subset of β which traverses almost-horizontal arcs with respect to q_{t}, we have |β_{0,t}|_{q_{t}} ≥ |β \ β_{0,t}|_{q_{t}}. Now by Lemma 5.3, we have t_{2} with bounded diam_{C}(F ([t_{1}, t_{2}])) such that |β|_{t_{2}} ≥ ϵ_{1}, and therefore |β_{0,t_{2}}|_{q_{t_{2}}} ≥ ϵ_{1}/2. It may still be that this length is obtained by traversing many times a very short almost horizontal curve σ, but applying Lemma 5.3 again we obtain t_{3} with bounded diam_{C}(F ([t_{2}, t_{3}])) such that |σ|_{q_{t_{3}}} ≥ ϵ_{1}.

In particular β contains an embedded straight segment τ which is almost-horizontal with respect to q_{t_{3}} and of length at least ϵ.

Now suppose β is not almost horizontal for q_{t_{3}}, so that there is a segment β_{1} of β which has length at least ϵ and is not almost horizontal. Since β has no self intersections, β_{1} is disjoint from τ. Applying Lemma 5.5, there exists t_{4} with diam_{C}(F ([t_{3}, t_{4}])) ≤ d_{3}, such that if t > t_{4} then either β_{1} is almost horizontal with respect to q_{t}, or |β_{1}|_{q_{t}} ≤ ϵ_{1}. In the latter case, set t_{5} = t_{4} + log 2ϵ_{1}/ϵθ, and note that for t > t_{5}, β_{1} will either have length less than ϵ or be almost horizontal. Apply this to all of the saddle connections of β.

The next lemma shows that, unless a curve is almost vertical in L_{q}(0), it can be balanced in the forward direction after an interval of bounded size in the curve complex.

Lemma 5.7. (Almost Vertical) If γ is not almost vertical with respect to q then for

J = {t ≥ 0 : |γ|_{q_{t},v} > |γ|_{q_{t},h}}

we have diam_{C}(J) ≤ d_{5} = d_{5}(ϵ, θ).

Proof. Since γ is not almost vertical with respect to q, it contains a segment τ that is not almost vertical and has length at least ϵ. For t_{1} = log 2/θ, τ will be almost horizontal and have length at least ϵ with respect to q_{t_{1}}. We can assume that there is a set of almost vertical saddle connections ω ⊂ γ that carry at least 1/2 of the length of γ, for otherwise |γ|_{q_{t},h} would dominate for t > t_{2} for a bounded t_{2}. Obviously each ω is disjoint from τ, since γ does not have self intersections.

By Lemma 5.5, there is t_{3} > t_{1} with diam_{C}(F ([t_{1}, t_{3}])) ≤ d_{3} such that either ω is almost horizontal or has length at most ϵ_{1} with respect to q_{t_{3}}. If all the ω’s are in the former case we are done since then γ would have been balanced for t ≤ t_{3}.

Suppose, then, that some ω has length smaller than ϵ_{1}. If we follow any strand of γ starting at ω until it returns with the same orientation, we may find in the set of saddle connections it traverses a geodesic loop homotopic to a simple loop. There is a bound (in terms of the number of possible saddle connections n_{2}) on the number of such loops γ′, and hence there must be some γ′ each of whose saddle connections are traversed at least a definite fraction of the number of times ω is traversed.

Either γ′ contains an almost horizontal saddle connection, or it has length at most n_{2}ϵ_{1}. In that case, within bounded t > t_{3} it will either have length ϵ_{1} or begin to grow, and we may apply Lemma 5.3 to get a t_{4} with bounded diam_{C}(F ([t_{3}, t_{4}])), such that for t > t_{4} |γ′|_{q_{t}} ≥ ϵ_{1} and |γ′|_{q_{t}} is growing, which implies that after an additional bounded t_{5}, it will be balanced.

Thus for t > t_{5}, the contribution of ω to γ is either itself horizontal or offset (to within a bounded factor) by the other segments in γ′, and after another bounded interval [t_{5}, t_{6}] we have balance.

5.2. Completion of proof. We must show that all three conditions of definition 2.2 hold for our projection π_{q}, with suitable constants a, b and c (independent of the geodesic L_{q}).

In what follows, fix q and x = L_{q}(0). Let α = F_{q}(0) be a shortest curve on x. Assume π_{q}(β) = 0 so that β is balanced at 0. We assume that all curves are q-geodesics (hence q_{t}-geodesics for any t), and furthermore that ϵ, θ satisfy the conditions in Lemma 4.8 and Lemma 5.5.

Let us restate the conditions in our current terminology, and prove them. Condition (1): diam_{C}(F_{q}([0, π_{q}(F_{q}(0))])) ≤ c.

We may assume |α|_{q,v} > |α|_{q,h}, or equivalently that the balance point π_{q}(α) is positive. Since α has minimal extremal length with respect to x, we have a bound |α|_{q} ≤ Ext_{x}(α)^{1/2} ≤ \sqrt{e_{0}}. Thus for bounded t_{1}, the vertical length |α|_{q_{t_{1}} ,v} becomes ϵ_{1}, so either α is balanced for t ≤ t_{1}, in which case we are done, or |α|_{q_{t_{1}}} ≤ ϵ_{1}. In the latter case we apply Lemma 5.3 to see that either α is vertical, in which case diam_{C}(F ([0, ∞))) is bounded and π_{q}(α) = +∞, so we are done, or there is t_{2} with diam_{C}(F ([t_{1}, t_{2}])) bounded, and |α|_{t} is increasing after t_{2}, so it is balanced for some t < t_{2} + \frac{1}{2} cosh^{−1} \sqrt{2} and again we are done.

Condition (2): If d_{C}(β, γ) ≤ 1 then diam_{C}(F_{q}([π_{q}(β), π_{q}(γ)])) ≤ c.

Recall π_{q}(β) = 0. Assume π_{q}(γ) > 0, so |γ|_{q,v} > |γ|_{q,h}. Since β is balanced at 0, Lemma 5.6 gives t_{1} > 0 with diam_{C}(F_{q}([0, t_{1}])) bounded, such that β is almost horizontal with respect to q_{t_{1}}. Lemma 5.3 then gives t_{2} with diam_{C}(F_{q}([t_{1}, t_{2}])) bounded, so that |β|_{q_{t_{2}}} ≥ ϵ_{1}. Thus there is a t_{3} ≥ t_{2} with F_{q}[t_{2}, t_{3}] bounded, so that β has an almost horizontal segment of length ϵ with respect to q_{t_{3}}.

Now, either γ is already balanced for t ≤ t_{3}, in which case we are done, or it is still mostly vertical with respect to q_{t_{3}}. In this case, since γ is disjoint from β it misses the horizontal segment of length ϵ and Lemma 5.5 gives t_{4} with diam_{C}(F_{q}([t_{3}, t_{4}])) bounded, so that every saddle connection of γ is either almost horizontal or has length at most ϵ_{1} with respect to q_{t_{4}}. Thus for t_{5} with bounded t_{5} − t_{4}, either the length of γ shrinks to ϵ_{1}, or it begins to increase so γ is balanced for t ≤ t_{5}. In the former case, Lemma 5.3 says that the segment J of t > t_{5} where |γ|_{q_{t}} ≤ ϵ_{1} has bounded-diameter image in C. This includes the case where γ is completely vertical and π_{q}(γ) = +∞. In all other cases, γ will be balanced for some t ∈ J and again we are done.

Condition (3): If d_{C}(β, F_{q}(π_{q}(β))) ≥ a and d_{C}(β, γ) ≤ bd_{C}(β, F_{q}(π_{q}(β))) then diam_{C} F_{q}([π_{q}(β), π_{q}(γ)]) ≤ c.

Recall that F_{q}(π_{q}(β)) = F_{q}(0) = α. Assume without loss of generality that γ is more vertical than horizontal at q_{0}. By Proposition 5.2, we can assume that β ∈ C_{b}, for otherwise its distance from the image of F is at most 1. By Lemma 5.6 (Almost Horizontal), there is some t_{1} > 0 with diam_{C}(F_{q}[0, t_{1}]) bounded such that β is almost horizontal at q_{t_{1}}.

We next show that γ cannot be almost vertical at q_{t_{1}}. Let M, h be the constants given by lemma 4.8, and suppose by contradiction that γ is almost vertical. If α′ is any curve of q_{t_{1}}-length at most M, there is a bound d(M) on d_{C}(α′, α) by the following: Lemma 4.1 gives a nonperipheral annulus A of width W so that the intersection of α′ with its core σ is at most M/W. Lemma 2.1 then bounds d_{C}(α′, σ). Lemma 2.5 in turn bounds d_{C}(σ, F_{q}(t_{1})) since both have bounded extremal length. Finally d_{C}(F_{q}(t_{1}), F_{q}(0)) = d_{C}(F_{q}(t_{1}), α) is bounded by choice of t_{1}.

Now applying Lemma 3.12 with Q = 2/h, and k = d(M), we obtain, provided d_{C}(α, β) ≥ D_{3} and d_{C}(γ, β) ≤ νd_{C}(α, β), that

min_{α′} i(β, α′) min_{α′} i(γ, α′) ≥ Qi(β, γ)

where α′ varies over all curves of q_{t_{1}}-length at most M. On the other hand, Lemma 4.8 gives the opposite inequality

i(γ, β) ≥ h min_{α′} i(β, α′) min_{α′} i(γ, α′).

This is a contradiction since we have chosen Q > 1/h, and we conclude that γ cannot be almost vertical at t_{1}. Thus by Lemma 5.7 (Almost Vertical), there exists t_{2} with diam_{C}(F_{q}([t_{1}, t_{2}])) bounded such that γ is balanced at t_{2}. This concludes the proof of Theorem 2.6.

6. Contraction property and hyperbolicity

To complete the proof of Theorem 1.1, it remains to prove Theorem 2.3, that if a geodesic metric space X has a coarsely transitive path family Γ with the contraction property then X is hyperbolic.

For our purposes a path γ : I → X is a quasi-geodesic if the following inequality holds for any x, y ∈ I:

length_{s}(γ[x, y]) ≤ Kd_{X}(γ(x), γ(y)) + δ

where K ≥ 1 and δ, s ≥ 0 are fixed constants, and length_{s} for s > 0 is “arclength on the scale s”, which is defined as follows: length_{s}(γ[x, y]) = sn where n is the smallest number for which [x, y] can be subdivided into n closed subintervals J_{1}, …, J_{n} with diam_{X}(γ(J_{i})) ≤ s. (This definition circumvents the need for checking the behavior of the parametrization at small scale; we let length_{0} denote normal length). Note also that the opposite inequality d_{X}(γ(x), γ(y)) ≤ length_{s}(γ[x, y]) holds automatically.

The proof is in two steps. We say that X has stability of quasi-geodesics if for all K ≥ 1,δ, s ≥ 0 there exists R > 0 such that any (K, δ, s)-quasi-geodesic α : I → X with endpoints x, y remains in an R-neighborhood of any geodesic [xy].

Lemma 6.1. If X has a coarsely transitive path family Γ with the contraction property then X has stability of quasi-geodesics. In addition, the paths of Γ themselves are quasi-geodesics.

Lemma 6.2. Stability of quasi-geodesics implies hyperbolicity.

Proof of Lemma 6.1. We may assume that the path family Γ is transitive, since for paths of length bounded by a fixed D it is easy to define a contracting projection, simply by mapping all of X to one endpoint.

Consider γ : [0, M] → X in Γ, and let α : [0, L] → X be a (K, δ, s)-quasi-geodesic such that α(0) = γ(0) and α(L) = γ(M). We show that α remains in a R(K, δ, s)- neighborhood of γ. The proof is somewhat complicated by the fact that we do not assume continuity of α, γ or π, but the idea is simple and well-known: large excursions of α away from γ can, using the contraction property, be circumvented by short cuts that travel along the projection to γ.

Let r(u) = d(α(u), γ(π(α(u)))). We will bound r(u) uniformly in terms of K, δ, s, and the constants a, b and c of the contraction property (Definition 2.2).

Divide [0, L] into closed intervals J_{1}, …, J_{n} such that ns = length_{s}(α[0, L]) and diam α(J_{i}) ≤ s. Then by part (2) of definition 2.2, diam γ(π(α(J_{i}))) ≤ s′, where s′ = c if s ≤ 1 and s′ = 1 + cs if s > 1.

Fix R_{0} > 0, to be determined shortly. For any u ∈ [0, L], if r(u) ≥ R_{0} + s′ then u is contained in some interval J = [u_{0}, u_{1}], a union of J_{i}, such that r ≥ R_{0} in J and r ≤ R_{0} + s′ at u_{0} and u_{1}. Subdivide J into intervals K_{1}, …, K_{m}, each a union of at most bR_{0}/s of the J_{i}, so that for each j diam α(K_{j}) ≤ bR_{0}, and the number m is at most 1 + length_{s}(α(J))/bR_{0}. Now assuming R_{0} ≥ a and applying the contraction property (part 3) to each of these we obtain

diam γ([π(α(u_{0})), π(α(u_{1}))]) ≤ mc

and by the triangle inequality

d(α(u_{0}), α(u_{1})) ≤ 2(R_{0} + s′) + (1 + \frac{length_{s}(α(J))}{bR_{0}}) c. (6.1)

Since α is a (K, δ, s)-quasi-geodesic, length_{s}(α(J)) ≤ Kd(α(u_{0}), α(u_{1})) + δ. Combining with (6.1), we get

(6.2) length_{s}(α(J)) ≤ \frac{Kc}{bR_{0}} length_{s}(α(J)) + 2Kc(R_{0} + s′) + δ.

Make the (a priori) choice of R_{0} sufficiently large that Kc/bR_{0} < 1/2. Then (6.2) gives an upper bound R on length_{s}(α(J)) depending only on the initial constants.

Thus d(α(u), {α(u_{0}), α(u_{1})}) is at most R/2, and in particular, no point in α(J) can be further than R_{0} + R/2 from γ([0, M]). Furthermore by applying part (2) of the contraction property, it follows that r(u) is bounded uniformly.

This implies that we can project from γ back to α, in the following sense: For any t ∈ [0, M] we can find u ∈ [0, L] such that d(γ(t), γ(π(α(u)))) is bounded by a uniform constant, just by chopping α into bounded-length pieces and applying parts (1) and (2) of the contraction property. Now by the bound on r(u) we can bound d(γ(t), α(u)) uniformly.

Apply this to an actual geodesic α and a quasi-geodesic β with the same endpoints. Letting γ ∈ Γ be a path with the same endpoints, we project from β to γ and then from γ to α as above. Both steps move a bounded distance, so we conclude that β lies in a bounded neighborhood of α. Hence, we have stability of quasi-geodesics.

Proof of Lemma 6.2. To prove hyperbolicity it suffices to establish the thin triangle condition. Let x, y, z be three points in X. We must show that [xy] lies in a δ-neighborhood of [xz] ∪ [yz], for uniform δ.

Let z′ ∈ [xy] be a point that minimizes distance from z to [xy]. We claim that the broken geodesic [xz′]∪[z′z] is a (3, 0, 0)-quasi-geodesic. If z′ = x this is obvious, so assume z′ ≠ x. Let u lie in [xz′] and v lie in [z′z].

It follows from the choice of z′ that it also minimizes distance from [v] to [xy] (via the triangle inequality). Thus d(u, v) ≥ d(z′, v).

By the triangle inequality, d(u, v) ≥ d(u, z′) − d(z′, v). Thus adding this to twice the previous inequality we get 3d(u, v) ≥ d(z′, v) + d(u, z′). This is exactly the fact that length([uz′] ∪ [z′v]) estimates d(u, v), so we conclude [xz′] ∪ [z′z] is a (3, 0, 0)- quasi-geodesic.

Now by stability of quasi-geodesics, we have that [xz′] ∪ [z′z] is in a uniform δ-neighborhood of [xz], and in particular [xz′] is. Applying the same argument for y replacing x, we see that all of [xy] is in a δ-neighborhood of [xz] ∪ [yz]. This concludes the proof.

7. Relative Hyperbolicity

In this final section we establish Theorems 1.2 and 1.3, which provide an interpretation of our hyperbolicity theorem in terms of the geometry of Teichmüller space, and the structure of the Mapping Class Group.

The following terminology is due to Farb [8]: If X is any geodesic metric space and H is a family of regions in X, let the electric distance d_{e} on X be the path metric imposed by shrinking each H ∈ H to diameter 1, in the following way: For each H ∈ H create a new point c_{H} and an interval of length 1/2 from c_{H} to every point in H. The new metric is induced by shortest paths in this enlarged space X̂ (called the electric space). We say X is relatively hyperbolic with respect to H if (X̂, d_{e}) is δ-hyperbolic for some δ.

7.1. In Teichmüller space. Fixing ϵ_{0} > 0 sufficiently small that the Collar Lemma holds for ϵ_{0}, let H_{C} = {H_{α}}_{α∈C_{0}(S)} denote the family of regions in T (S) defined as in the introduction:

H_{α} = {x ∈ T (S) : Ext_{x}(α) < ϵ_{0}}.

Then it is easy to see that a set of points α_{1}, …, α_{k} is a simplex in C(S) if and only if H_{α_{1}} ∩ ⋯ ∩ H_{α_{k}} is non-empty. In other words, C(S) is the nerve of the family H_{C}.

The statement of Theorem 1.2 is a direct consequence of Theorem 1.1 and the following:

Lemma 7.1. The electric space (T̂(S), d_{e}) defined with respect to the family H_{C} is quasi-isometric to C_{1}(S).

Proof. There is a natural map φ : C_{0}(S) → T̂(S) taking each α to the new point c_{α} ≡ c_{H_{α}}. The set C_{0}(S) is clearly 1/2-dense in C_{1}(S). Let us check that its image {c_{α}} is d_{0}-dense in T̂(S), for some d_{0} < ∞.

Recall that for any conformal structure x on S there is a curve α ∈ C_{0}(S) with Ext_{x}(α) ≤ e_{0}. Then for this α, we see that x is a bounded Teichmüller distance (in fact \frac{1}{2} log(e_{0}/ϵ_{0})) from H_{α}: we may apply to x a Teichmüller map whose vertical foliation consists of leaves homotopic to α. It follows that \{c_{α}\} is (\frac{1}{2} + \frac{1}{2} log(e_{0}/ϵ_{0}))- dense in T̂(S).

Now we need only show that for any α, β ∈ C_{0}(S)

\frac{1}{K} d_{C}(α, β) − a ≤ d_{e}(c_{α}, c_{β}) ≤ Kd_{C}(α, β) + a (7.1)

with fixed K, a > 0, to show that φ induces a quasi-isometry. One direction is easy: if d_{C}(α, β) = 1 then H_{α} ∩ H_{β} is nonempty, and any point x in this set is connected to each of c_{α} and c_{β} by a segment of length 1/2. Hence φ is 1-Lipschitz.

To obtain the other direction, consider for any x ∈ T (S) the set Φ(x) of elements in C_{0}(S) of minimal Ext_{x}. This set has diameter at most 2e_{0} + 1 by Lemma 2.4.

Now if d_{T} (S)(x, y) ≤ 1 we see also that Φ(x) ∪ Φ(y) has bounded diameter, by Lemma 5.1. Note also that if x ∈ H_{α} then d_{C}(α, Φ(x)) ≤ 1.

Thus, any map that associates to x ∈ T (S) some (any) element of Φ(x) and to c_{α} associates α will expand distances by a bounded multiplicative and additive amount, and serve as an inverse to φ. It follows that φ is a quasi-isometry.

7.2. In the Mapping Class Group. To carry out a similar analysis for Mod(S), recall first that for any group G with a fixed finite generating set Γ, the Cayley graph G = G_{G,Γ} is a 1-complex whose vertex set is G and whose edges are all pairs (g, gγ) with γ ∈ Γ. Giving all edges length 1, we obtain a complete locally finite geodesic metric space.

Now for G = Mod(S), we single out a number of subgroups as follows. Up to the action of Mod(S), there are only a finite number of distinct non-trivial nonperipheral homotopy classes of simple curves in S (distinguished by the topological type of their complement). Let {α_{1}, …, α_{N}} be a fixed list of representatives of these Mod(S)-orbits. Let F ix(α_{j}) be the subgroup of Mod(S) fixing α_{j}.

Given any β ∈ C_{0}(S), let α_{j} be the unique representative of β in the list, and let G_{β} be the left-coset of F ix(α_{j}) defined by G_{β} = {g ∈ Mod(S) : g(α_{j}) = β}.

Now we may form the electric space Ĝ of G relative to the family of cosets {G_{β}}, and its electric distance d_{e}. The analogue to Lemma 7.1 is:

Lemma 7.2. Fixing a choice of generating set Γ and representatives {α_{1}, …, α_{N}} of Mod(S)-orbits in C_{0}(S), the electric space (Ĝ, d_{e}) is quasi-isometric to C_{1}(S).

Again, this together with Theorem 1.1 proves Theorem 1.3, where the relative hyperbolicity of Mod(S) is with respect to this family of cosets {G_{β}}.

Proof. Let c_{β} denote the new point added to G_{β} in the construction of Ĝ. The natural map φ : C_{0}(S) → Ĝ is again φ(β) = c_{β}. In this case it is clear that {c_{β}} is 1/2-dense in Ĝ since every g ∈ Mod(S) is in the coset gF ix(α_{j}) = G_{g(α_{j})} for each j ≤ N. It remains to check that the inequalities (7.1) hold.

Up to the action of Mod(S) there are only finitely many pairs (β, β′) of disjoint curves in C_{0}(S) (i.e. edges in C_{1}(S)). Let {(β_{i}, β′_{i})}_{i=1}^{L} be an enumeration of representatives of Mod(S)-orbits. For each α_{j} there is some (in fact several) β_{i} equivalent to it under Mod(S), so let w_{ij} be a fixed group element such that w_{ij}(α_{j}) = β_{i}. Define w′_{ij} similarly. Since this is a finite list, there is some upper bound B on their lengths as words in the generating set Γ.

Now let β, β′ ∈ C_{0}(S) be any two curves of distance 1. Hence there exists g ∈ Mod(S) and i ≤ L such that g(β_{i}) = β and g(β′_{i}) = β′. There also exist j, k ≤ N such that w_{ij}(α_{j}) = β_{i} and w′_{ik}(β_{k}) = β′_{i}.

Thus gw_{ij} ∈ G_{β} and gw′_{ik} ∈ G_{β′}, and these two elements are separated by a path in G of distance at most 2B. We conclude that d_{e}(c_{β}, c_{β′}) ≤ 2B + 1 and hence the map φ is (2B + 1)-Lipschitz.

To obtain a bound in the other direction, note that for any g ∈ Mod(S) we may associate the set A_{g} = {g(α_{j})}_{j≤N} in C_{0}(S), and that the diameter of this set in C(S) is equal to the diameter of A_{id} = {α_{j}}_{j≤N}, which is some fixed D (with appropriate choice of α_{J} we can easily get D = 2). Now given g and gγ where γ ∈ Γ is a generator, the distance between the sets A_{g} and A_{gγ} is equal to that between A_{id} and A_{γ}, which is again bounded. Note finally that if g ∈ G_{β} then β ∈ A_{g}. Thus we can map the vertices of Ĝ back to C_{0}, taking each c_{β} to β, and each g to some (any) element of A_{g}, and the resulting map is Lipschitz, and inverts φ. This proves that φ is a quasi-isometry.

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