Contemporary Mathematics

Unstable quasi-geodesics in Teichmüller space

Howard A. Masur and Yair N. Minsky

In this note we will construct an example of a quasi-geodesic in the Teichmüller space of a surface with the following instability property:

Theorem 0.1. For any hyperbolic surface S whose Teichmüller space T (S) has dimension at least 4, there is a bi-infinite quasi-geodesic L in T (S) such that

1. L projects to a compact part of the moduli space M(S), but

2. L does not lie in a bounded neighborhood of any geodesic.

(Recall that a quasi-geodesic, or (K, ϵ)-quasi-geodesic for K > 1, ϵ > 0, is a path for which the length ℓ and distance d between endpoints of any subsegment satisfy ℓ ≤ Kd + ϵ.)

This theorem answers in the negative a question asked us by M. Mitra (see [7] for a related sharper question), and should be contrasted with Theorem 4.2 from [6], which gives the following stability property: Given a fixed compact set C ⊂ M(S) and K > 1, ϵ > 0, there is a constant R so that, if G is any geodesic whose projection to M(S) lies in C, and if L is any (K, ϵ)-quasi-geodesic joining two points of G, then L lies in an R-neighborhood of G. This stability property holds, by comparison, in hyperbolic spaces for all geodesics G, but in Teichmüller space it fails if G does not project to a compact set.

This suggests that “in the thick part of Teichmüller space things are essentially hyperbolic”. Our theorem, on the other hand, shows that the roles of the geodesic and the quasi-geodesic cannot be reversed, even in the thick part, and hence that this stability is somewhat delicate.

Note of course that when dim(T (S)) = 2 the theorem is false: T (S) is then the hyperbolic plane, in which all quasi-geodesics are uniformly close to geodesics.

We refer the reader to [4] for basic notions of Teichmüller theory; to [2] and [3] for material on Thurston’s measured lamination (or foliation) spaces and on pseudo-Anosov homeomorphisms; and to [1] for more on quasi-geodesics and their stability properties in hyperbolic spaces.

Proof

Fix any essential proper subdomain Y of S, other than an annulus or a thricepunctured sphere. By “essential” we mean that π_{1}(Y) injects into π_{1}(S). Let h : S → S be a homeomorphism which preserves Y, is the identity outside Y, and whose restriction to Y is in a pseudo-Anosov mapping class. The action of h on T (S) takes a conformal structure x to its pullback by h^{−1}.

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Fix x_{0} ∈ T (S) and let x_{n} = h^{n}(x_{0}). Join x_{0} to x_{1} by a fixed path (say geodesic) p_{0}, let p_{n} = h^{n}(p_{0}), and let L denote the infinite concatenation ⋯ ∗ p_{−2} ∗ p_{−1} ∗ p_{0} ∗ p_{1} ∗ p_{2} ∗ · · ·.

Clearly L projects to a compact subset of M(S), namely the image of p_{0}. We first show that L is a (K, ϵ)-quasi-geodesic, for some fixed K > 1 and ϵ > 0.

Since each segment p_{n} has fixed length |p_{0}|, and since L is invariant under h, it suffices to show for some C > 0 that d(x_{0}, x_{n}) ≥ Cn for all n > 0. Let μ be the unstable lamination of h in its action on Y. Then there exists λ > 1 such that h^{−n}(μ) = λ^{n}μ. Let ML(S) denote Thurston’s space of measured laminations on S. For any measured lamination ν let E_{x}(ν) denote the extremal length of ν on x ∈ T (S) – this was defined for all measured laminations by Kerckhoff [5] and generalizes the extremal length of a curve. In particular E_{x} is homogeneous of degree 2 in lamination space. Then

E_{x_{n}}(μ) = E_{x_{0}}(h^{−n}(μ)) = λ^{2n}E_{x_{0}}(μ).

Kerckhoff also established in [5] that Teichmüller distance can be given as:

d(x, y) = \frac{1}{2} log sup_{ν∈ML(S)} \frac{E_{y}(ν)}{E_{x}(ν)}.

It follows immediately that

d(x_{0}, x_{n}) ≥ n log λ.

This establishes that L is a quasi-geodesic.

Now we show that L cannot stay in a bounded neighborhood of any geodesic. Fix M > 0 and let y_{−n} and y_{n} be any two points within M of x_{−n} and x_{n} respectively. Let G_{n} be the Teichmüller geodesic joining y_{−n} to y_{n}. Let Φ_{n} be the unit-area quadratic differential on y_{−n} associated to G_{n}. Let α be a boundary curve of Y.

We first claim that, for any unit-area quadratic differential Φ on any y_{i} within distance M of x_{i}, ℓ_{Φ}(α) is bounded uniformly above and below by positive constants, depending on M, where ℓ_{Φ} denotes minimal length of a homotopy class in the singular flat metric |Φ|. Indeed, if Φ is any unit-area quadratic differential on a surface y_{0} within distance M of x_{0} then ℓ_{Φ}(α) is bounded above and below by positive constants, just by compactness of the set of unit-area quadratic differentials on a compact set of surfaces in Teichmüller space. Transporting y_{0} to y_{i} by h^{i} and noting that h(α) = α, we obtain the desired estimate.

If ℓ_{Φ,h} and ℓ_{Φ,v} denote lengths in the horizontal and vertical directions of Φ respectively, then both of these are also bounded above for α by the previous estimate. Apply this to Φ_{n} on y_{−n}, and then to its image on y_{n} by the Teichmüller flow along G_{n}, which we call Φ′_{n}. On both of these we then get uniform upper bounds on the horizontal and vertical lengths of α. However, ℓ_{Φ_{n},h} increases exponentially along G_{n}, and ℓ_{Φ_{n},v} decreases exponentially. It follows that on the midpoint m_{n} of G_{n}, both are bounded above by ce^{−n} for a fixed c.

Thus the length of α on m_{n} with respect to the unit-area differential is going to zero as n → ∞. This means m_{n} has to leave every compact set of the moduli space. For in any compact set of the moduli space there is a lower bound on ℓ_{Φ} over all unit-area quadratic differentials and simple closed curves.

It follows that G_{n} cannot remain in a bounded neighborhood of L for all n. Now, if L were in a bounded neighborhood of some geodesic G then one easily sees that G would be in a bounded neighborhood of L. An exhaustion of G by subsegments G_{n} would then lead to a contradiction. This establishes the theorem.

References

[1] J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, Notes on word hyperbolic groups, Group Theory from a Geometrical Viewpoint, ICTP Trieste 1990 (E. Ghys, A. Haefliger, and A. Verjovsky, eds.), World Scientific, 1991, pp. 3–63.

[2] A. J. Casson and S. A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, Cambridge University Press, 1988.

[3] A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces, vol. 66-67, Asterisque, 1979.

[4] F. Gardiner, Teichmüller theory and quadratic differentials, Wiley Interscience, 1987.

[5] S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), 23–41.

[6] Y. Minsky, Quasi-projections in Teichmüller space, J. Reine Angew. Math. 473 (1996), 121– 136.

[7] Mahan Mitra, Coarse extrinsic geometry: a survey, The Epstein birthday schrift, Geom. Topol., Coventry, 1998, pp. 341–364 (electronic).

University of Illinois at Chicago

    E-mail address: masur@math.uic.edu

SUNY at Stony Brook

    E-mail address: yair@math.sunysb.edu