Groups of homeomorphisms of one-manifolds, III: Nilpotent subgroups

Benson Farb^{∗} and John Franks^{†}

August 14, 2001

Abstract

This self-contained paper is part of a series [FF1, FF2] seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. Plante-Thurston proved that every nilpotent subgroup of Diff^{2}(S^{1}) is abelian. One of our main results is a sharp converse: Diff^{1}(S^{1}) contains every finitely-generated, torsion-free nilpotent group.

1 Introduction

A basic aspect of the theory of linear groups is the structure of nilpotent groups. In this paper we consider nilpotent subgroups of Homeo(M) and Diff^{r}(M), where M is the line R, the circle S^{1}, or the interval I = [0, 1]. As we will see, the structure theory depends dramatically on the degree r of regularity as well as on the topology of M.

Throughout this paper all homeomorphisms will be orientation-preserving, and all groups will consist only of such homeomorphisms. Plante-Thurston [PT] discovered that C^{2} regularity imposes a severe restriction on nilpotent groups of diffeomorphisms.

Theorem 1.1. Any nilpotent subgroup of Diff^{2}(I), Diff^{2}([0, 1)) or Diff^{2}(S^{1}) must be abelian.

Remark. In the case of subgroups of Diff^{2}(I) and Diff^{2}([0, 1)) this result was first proved by Plante and Thurston in [PT]. They also proved that the group is virtually abelian in the S^{1} case. For completeness of exposition (and because it is simple) we present a proof of the Plante-Thurston result about I. These results are related to a result of E. Ghys [Gh], who proved that any solvable subgroup of Diff^{ω}(S^{1}) is metabelian. In §4 we prove the PL version of Theorem 1.1.

∗Supported in part by the NSF and by the Sloan Foundation.

†Supported in part by NSF grants DMS9803346 and DMS0099640.

Our main result is that lowering the regularity from C^{2} to C^{1} produces a sharply contrasting situation, where every possibility can occur.

Theorem 1.2. Let M = R, S^{1}, or I. Then every finitely-generated, torsion-free nilpotent group is isomorphic to a subgroup of Diff^{1}(M).

Witte has observed ([Wi], Lemma 2.2) that the groups of orientation-preserving homeomorphisms of R are precisely the right-orderable groups. Since torsionfree nilpotent groups are right-orderable (see, e.g. [MR], p.37), it follows that such groups are subgroups of Homeo(R). The content of Theorem 1.2 is the increase in regularity from C^{0} to the sharp regularity C^{1}.

The situation for R is more complicated. On the one hand there is no limit to the degree of nilpotence, even when regularity is high:

Theorem 1.3. Diff^{∞}(R) contains nilpotent subgroups of every degree of nilpotency.

On the other hand, even with just a little regularity, the derived length of nilpotent groups is greatly restricted.

Theorem 1.4. Every nilpotent subgroup of Diff^{2}(R) is metabelian, i.e. has abelian commutator subgroup.

In particular, Theorem 1.4 gives that the group of n × n upper-triangular with ones on the diagonal, while admitting an effective action on R by C^{1} diffeomorphisms, admits no effective C^{2} action on R if n > 3.

Closely related to this algebraic restriction on nilpotent groups which act smoothly is a topological restriction.

Theorem 1.5. If N is a nilpotent subgroup of Diff^{2}(R) and every element of N has a fixed point then N is abelian.

One open problem is to extend the above theory to solvable subgroups, as well as to higher-dimensional manifolds. The paper of J. Plante [P] contains a number of interesting results and examples concerning solvable groups acting on R. Another problem is to understand what happens between the degrees of regularity r = 0 and r ≥ 2, where vastly differing phenomena occur.

Residually nilpotent groups. A variation of our construction of actions of nilpotent groups can be used to construct actions of a much wider class of groups, the residually torsion-free nilpotent groups, i.e. those groups where the intersection of all terms in the lower central series is trivial. In §2.4 we prove the following result.

Theorem 1.6. Let M = R, S^{1} or I. Then Diff_{+}^{1}(M) contains every finitely generated, residually torsion-free nilpotent group.

The class of finitely generated, residually torsion-free nilpotent groups includes free groups, surface groups, and the Torelli groups, as well as products of these groups.

Acknowledgements. It is a pleasure to thank Dave Witte for making several useful suggestions, in particular pointing us to the references [Ra], [P], and [MR].

2 Constructions

In this section we construct examples of nilpotent groups acting on one-manifolds.

2.1 An elementary observation

There several obvious relations among actions on the three spaces R, S^{1}, and I. Restricting an action on I to the interior of I gives an action on R. One can also start with an action on R and consider the action on the one-point compactification S^{1} or the two-point compactification I. But this usually entails loss of regularity of the action. That is, a smooth or PL action on R will generally give only an action by homeomorpisms on I or T^{1}.

2.2 C^{1} actions on M

Let N_{n} denote the group of n × n lower-triangular integer matrices with ones on the diagonal. Our first main goal is to prove that N_{n} admits C^{1} actions on the real line.

The proof is somewhat technical and so we will describe the strategy before engaging in the details. The traditional proof that N_{n} acts by homeomorphisms on the interval uses the fact that it is an ordered group. Suppose a group acts effectively on a countable ordered set in a way that preserves the order (e.g. the set might be the group itself), One can then produce an action by homeomorphisms on I by embedding the countable set in an order preserving way in I and canonically extending the action of each group element on that set to an order preserving homeomorphism of I, first by using continuity to extend to the closure of the embedded set and then using affine extensions on the complementary intervals of this closure.

The approach we take is somewhat similar. We consider the group Z^{n} of ntuples of integers and provide it with a linear order ≻ which is the lexicographic ordering, i.e. (x_{1}, …, x_{n}) ≺ (y_{1}, …, y_{n}) if and only if x_{i} = y_{i} for 1 ≤ i < k and x_{k} < y_{k} for some 0 ≤ k ≤ n. It is well known and easy to show that the standard linear action of N_{n} on Z^{n} preserves this ordering.

Instead of embedding Z^{n} as a countable set of points, however, for each (q_{1}, …, q_{n}) ∈ Z^{n} we will embed a closed interval I(q_{1}, …, q_{n}) in I. We do this in such a way that the intervals are disjoint except that each one intersects its successor in a common endpoint, and so that the order of the intervals in I agrees with the lexicographic ordering on Z^{n}. We also arrange that the complement of the union of these intervals is a countable set. If for each α ∈ N_{n} we defined g_{α} on I(q_{1}, …, q_{n}) to be the canonical affine map to I(α(q_{1}, …, q_{n})) this would define g_{α} on a dense subset of I and it would extend uniquely to a homeomorphism, giving an action of N_{n} by homeomorphisms.

In order to improve this to a C^{1} action we must replace the affine maps from I(q_{1}, …, q_{n}) to I(α(q_{1}, …, q_{n})) with elements of another canonical family of diffeomorphisms which has two key properties. The first is that these subinterval diffeomorphisms must have derivative 1 at both endpoints in order to fit together in a C^{1} map. The second is that if I_{k} is a strictly monotonic sequence of these subintervals then the restriction of g_{α} to I_{k} must have a derivative which converges uniformly to 1 as k tends to infinity. In particular, by the mean value theorem the ratio of the lengths of α(I_{k}) and I_{k} must tend to 1 for any element α ∈ N_{n}. This makes the choice of the lengths of these intervals one of the key ingredients of the proof. We choose these lengths with a positive parameter K. We are then able to show that as this parameter increases to infinity the C^{1} size of any g_{α} goes to zero. This allows us to prove the existence of an effective C^{1} action of N_{n} on I with generators chosen from an arbitrary C^{1} neighborhood of the identity.

We proceed to the details beginning with some calculus lemmas. The first of them develops a family of interval diffeomorphisms which we will us as a replacement for affine interval maps as we discussed above. We are indebted to Jean-Christophe Yoccoz for the proof of the following lemma which substantially simplifies our earlier approach.

Lemma 2.1. For each a and b ∈ (0, ∞) there exists a C^{1} orientation preserving diffeomorphism ϕ_{a,b} : [0, a] → [0, b] with the following properties:

1. For any a, b, c ∈ (0, ∞) and for any x ∈ [0, a], ϕ_{b,c}(ϕ_{a,b}(x)) = ϕ_{a,c}(x).

2. For all a, b, ϕ′_{a,b}(0) = ϕ′_{a,b}(1) = 1.

3. Given ε > 0 there exists δ > 0 such that for all x ∈ [0, a],

|ϕ′_{a,b}(x) − 1| < ε, whenever |\frac{b}{a} − 1| < δ

Proof. For u_{0} ∈ (0, ∞) we define

L(u_{0}) = ∫_{−∞}^{∞} \frac{du}{u^{2} + u_{0}^{2}} .

A change of variables u = u_{0}x allows one to conclude that

L(v) = \frac{1}{u_{0}} ∫_{−∞}^{∞} \frac{dx}{x^{2} + 1} = \frac{π}{u_{0}}

For each u_{0} ∈ (0, ∞) we define ψ_{u_{0}} : R → (0, L(u_{0})) by

ψ_{u_{0}}(t) = ∫_{−∞}^{t} \frac{du}{u^{2} + u_{0}^{2}}

Clearly ψ_{u_{0}} is a real analytic diffeomorphism from R onto the open interval (0, L(u_{0})).

Now define ϕ_{a,b} : [0, a] → [0, b] by setting ϕ_{a,b}(0) = 0, ϕ_{a,b}(a) = b and for x ∈ (a, b) letting ϕ_{a,b}(x) = ψ_{u_{1}}(ψ_{u_{0}}^{−1}(x)) where a = L(u_{0}) and b = L(u_{1}). This is clearly a homeomorphism of [0, a] to [0, b] which is real analytic on (0, a). We observe that the derivative of ϕ_{a,b}(x) at the point x = ψ_{u_{0}}(t) is given by

ϕ′_{a,b}(x) = \frac{ψ′_{u_{1}}(t)}{ψ′_{u_{0}}(t)} = \frac{t^{2} + u_{0}^{2}}{t^{2} + u_{1}^{2}}

From this it is clear that ϕ′_{a,b}(x) can be continuously extended to the endpoints of the interval [0, a] by assigning it the value 1 there. Hence ϕ_{a,b}(x) is C^{1} on the closed interval [0, a] and satisfies property (2) above. Property (1) is clear from the definition.

To check property (3) we observe that if, as above x = ψ_{u_{0}}(t) then

|ϕ′_{a,b}(x) − 1| = |\frac{t^{2} + u_{0}^{2}}{t^{2} + u_{1}^{2}} − 1|

which is easily seen to assume its maximum when t = 0. But

|ϕ′_{a,b}(x) − 1| ≤ |\frac{u_{0}^{2}}{u_{1}^{2}} − 1| = |\frac{b^{2}}{a^{2}} − 1|

since u_{0} = π/a and u_{1} = π/b. From this it is clear that property (3) holds. ⋄

We will also need the following technical lemma.

Lemma 2.2. Suppose n is a positive even integer and K > 0. Then for (x, y) ∈ R^{2}

lim_{∥(x,y)∥→∞} \frac{|(x + y)^{n} − y^{n}|}{x^{n+2} + y^{n} + K} = 0.

Moreover, given ε > 0, for K sufficiently large

\frac{|(x + y)^{n} − y^{n}|}{x^{n+2} + y^{n} + K} < ε

for all (x, y) ∈ R^{2}.

Proof. The numerator (x + y)^{n} − y^{n} is a sum of mononomials of the form Cx^{k}y^{n−k} where 0 < k ≤ n. Hence it suffices to prove

lim_{∥(x,y)∥→∞} \frac{|x|^{k}|y|^{n−k}}{x^{n+2} + y^{n} + K} = 0

for 0 < k ≤ n.

If for some ε > 0

\frac{|x|^{k}|y|^{n−k}}{x^{n+2} + y^{n} + K} ≥ ε

we want to show that there is an upper bound for |x| and |y|.

We first observe that

\frac{|x|^{k}}{|y|^{k}} = \frac{|x|^{k}|y|^{n−k}}{|y|^{n}} > \frac{|x|^{k}|y|^{n−k}}{x^{n+2} + y^{n} + K} ≥ ε

or

|x| > ε^{1/k}|y|  (1)

Similarly,

\frac{|y|^{n−k}}{|x|^{n−k+2}} = \frac{|x|^{k}|y|^{n−k}}{|x|^{n+2}} > \frac{|x|^{k}|y|^{n−k}}{x^{n+2} + y^{n} + K} ≥ ε

This implies that

|y|^{n−k} > ε|x|^{n−k+2}  (2)

and we note that n−k +2 > 0. In the case that k = n we note that equation (2) implies that |x| is bounded and then equation (1) implies that |y| is bounded.

On the other hand, when k < n we have

|y| > E|x|^{d}  (3)

where

E = ε^{\frac{1}{n−k}} and d = \frac{n − k + 2}{n − k} = 1 + \frac{2}{n − k} .

Combining equations (1) and (3) we see

|x| > ε^{1/k}|y| > ε^{1/k}E|x|^{d}

Since d > 1 this clearly implies that |x| is bounded by a constant depending only on n and ε. Then equation (1) implies that |y| is also bounded. The contrapositive of these assertions is that for ∥(x, y)∥ sufficiently large we have

\frac{|x|^{k}|y|^{n−k}}{x^{n+2} + y^{n} + K} < ε

which implies the desired limit.

To prove the second assertion of the lemma we observe that we have shown there is a constant M > 0 such that

\frac{(x + y)^{n} − y^{n}}{x^{n+2} + y^{n} + K} < ε

whenever ∥(x, y)∥ > M. Morever the constant M is independent of K. Hence clearly if K is sufficiently large, this inequality will hold for all (x, y). ⋄

Consider the group Z^{n} of n-tuples of integers and provide it with a linear order ≻ which is the lexicographic ordering, i.e. (x_{1}, …, x_{n}) ≺ (y_{1}, …, y_{n}) if and only if x_{i} = y_{i} for 1 ≤ i < k and x_{k} < y_{k} for some 0 ≤ k ≤ n. We are now prepared to define a set of closed intervals in one-to-one correspondence with elements of Z^{n}. As mentioned above we will make this definition with a positive parameter K which will be used subsequently to get generators of our action in an arbitrary C^{1} neighborhood of the identity.

Definition 2.3. For K > 0 let B_{K} : Z^{n} → R be defined by

B_{K}(q_{1}, q_{2}, …, q_{n}) = K + ∑_{j=1}^{n} q_{j}^{2n−2j+2}
= q_{1}^{2n} + q_{2}^{2n−2} + ⋯ + q_{n−1}^{4} + q_{n}^{2} + K,

and let the constant S_{K} be defined by

S_{K} = ∑_{(q_{1},q_{2},...,q_{n})∈Z^{n}} \frac{1}{B_{K}(q_{1}, q_{2}, …, q_{n})}.

For (r_{1}, r_{2}, …, r_{n}) we define S_{K}(r_{1}, r_{2}, …, r_{n}) by

S_{K}(r_{1}, r_{2}, …, r_{n}) = ∑_{(q_{1},q_{2},...,q_{n})≺(r_{1},r_{2},...,r_{n})} \frac{1}{B_{K}(q_{1}, q_{2}, …, q_{n})}.

Finally we define the closed interval

I_{K}(r_{1}, r_{2}, …, r_{n}) = [S_{K}(r_{1}, r_{2}, …, r_{n}), S_{K}(r_{1}, r_{2}, …, r_{n} + 1)].

Note that the sum defining S_{K} converges, for example by the integral and comparison tests, so that S_{K} is well-defined. We will denote the interval [0, S_{K}] by I_{K} and we note that it is nearly the union of the intervals I_{K}(r_{1}, r_{2}, …, r_{n}).

Lemma 2.4. There is a countable closed set J_{K} in the interval I_{K} = [0, S_{K}], such that

I_{K} = J_{K} ∪ ( ⋃_{(q_{1},q_{2},...,q_{n})∈Z^{n}} I_{K}(q_{1}, q_{2}, …, q_{n}))

Proof. Define J_{K} by the equality in the statement of the lemma. From the definitions it follows that every point J_{k} is the limit of one of the Cauchy sequences {S_{K}(r_{1}, …, r_{i}, … r_{n})} as r_{i} ranges from 1 to either ∞ or −∞ and the other r_{j}, j ≠ i are fixed. It follows that J_{K} is countable. The set J_{K} is closed since its complement is clearly open. ⋄

We note that the intervals {I_{K}(q_{1}, q_{2}, …, q_{n})} occur in the interval I_{K} in exactly the order given by ≺ . They have disjoint interiors but each such interval intersects its successor in a common endpoint.

Definition 2.5. Let ν : I_{K}\J_{K} → Z^{n} be defined by setting ν(x) = (q_{1}, q_{2}, …, q_{n}) if x ∈ I_{K}(q_{1}, q_{2}, …, q_{n}) and x ∉ I_{K}(q_{1}, q_{2}, …, q_{n+1}).

Lemma 2.6. If K is sufficiently large, then for every α ∈ N_{n} there exists a homeomorphism g_{α} : I_{K} → I_{K} such that for each (q_{1}, …, q_{n}) ∈ Z^{n},

1. g_{α}(I_{K}(q_{1}, …, q_{n})) = I_{K}(α(q_{1}, …, q_{n})), and

2. g_{αβ}(x) = g_{α}(g_{β}(x)) for all α, β ∈ N_{n} and all x ∈ I_{K}.

3. for x ∈ I_{K}(q_{1}, …, q_{n}), g_{α}(x) = ϕ_{b_{1},b_{2}}(x − c_{1}) + c_{2}, where

b_{1} = |I_{K}(q_{1}, …, q_{n})| = B_{K}(q_{1}, …, q_{n})^{−1},
b_{2} = |I_{K}(α(q_{1}, …, q_{n})| = B_{K}(α(q_{1}, …, q_{n}))^{−1},
c_{1} = S_{K}(q_{1}, …, q_{n}), and
c_{2} = S_{K}(α(q_{1}, …, q_{n})).

and ϕ_{b_{1},b_{2}} : [0, b_{1}] → [0, b_{2}] is the diffeomorphism guaranteed by Lemma 2.1.

Proof. For x ∈ I_{K}(q_{1}, …, q_{n}), we define g_{α}(x) to be ϕ_{b_{2}}(ϕ_{b_{1}}^{−1}(x − c_{1})) + c_{2}. The values of the constants b_{j}, c_{j}, for j = 1, 2 were chosen so that (1) holds. This gives a homeomorphism g_{α} from I_{K} \ J_{K} to itself which preserves the order on the interval I_{K}. Since J_{K} is a closed countable subset of I_{K} we may extend g_{α} to all of I_{K} in the unique way which makes it an order preserving function on I_{K}. Clearly this makes g_{α} a homeomorphism of I_{K} to itself.

In order to show property (2) we let b_{3} = B_{K}(βα(q_{1}, …, q_{n}))^{−1} and c_{3} = S_{K}(βα(q_{1}, …, q_{n})). Then if x ∈ I_{K}(q_{1}, …, q_{n}),

g_{β}(g_{α}(x)) = ϕ_{b_{2},b_{3}}(g_{α}(x) − c_{2}) + c_{3}
= ϕ_{b_{2},b_{3}}(ϕ_{b_{1},b_{2}}((x − c_{1}) + c_{2} − c_{2}) + c_{3}, by Lemma 2.1
= ϕ_{b_{1},b_{3}}(x − c_{1}) + c_{3}
= g_{βα}(x).

Since this holds for any (q_{1}, …, q_{n}) ∈ Z^{n} we have shown g_{β}(g_{α}(x)) = g_{βα}(x) for a dense set of x ∈ I_{K}. Continuity then implies this holds for all x ∈ I_{K}. ⋄

Definition 2.7. For 1 ≤ i < n let σ_{i} ∈ N_{n} denote the matrix with all entries on the diagonal equal to 1, with entry (i + 1, i) equal to 1, and with all other entries 0. We will denote the homeomorphism g_{σ_{i}} : I_{K} → I_{K} by g_{i}.

The elements {σ_{i}} form a set of generators for the group N_{n}. Our next objective is to show that in fact g_{i} = g_{σ_{i}} is a C^{1} diffeomorphism. For this we will need a sequence of technical lemmas.

Lemma 2.8. Suppose x_{k} is a monotonic sequence in I_{K} \ J_{K} converging to a point of J_{K}. Then for each 1 ≤ i < n

lim_{k→∞} \frac{B_{K}(σ_{i}(ν(x_{k})))}{B_{K}(ν(x_{k}))} = 1.

Proof. We may assume without loss of generality that the sequence {x_{k}} is monotonic increasing. For 1 ≤ j ≤ n, k > 0 we define q_{j}(k) by ν(x_{k}) = (q_{1}(k), q_{2}(k), …, q_{n}(k)). Then each sequence {q_{j}(k)}_{k=1}^{∞} is monotonic increasing. At least one of these sequences is unbounded, since any bounded ones are eventually constant and if {q_{j}(k)} were eventually constant for all j ≤ n then the sequence of intervals I_{K}(ν(x_{k})) would be eventually constant and the limit of the sequence {x_{k}} would be in the final interval and hence not in J_{K}. Let r be the smallest j such that {q_{j}(k)} is unbounded. Then since {x_{k}} is monotonic increasing we have lim_{k→∞} q_{r}(k) = ∞ and for j < r the sequence {q_{j}(k)} is eventually constant.

We note that

B_{K}(σ_{i}(ν(x_{k}))) = B_{K}(q_{1}(k), …, q_{i}(k), q_{i+1}(k) + q_{i}(k), q_{i+2}(k), …, q_{n}(k))

and hence that

\frac{B_{K}(ν(x_{k})) − B_{K}(σ_{i}(ν(x_{k})))}{B_{K}(ν(x_{k}))} = \frac{(q_{i+1}(k) + q_{i}(k))^{2n−2i} − q_{i+1}(k)^{2n−2i}}{B_{K}(q_{1}(k), … q_{n}(k))}

= \frac{P (q_{i}(k), q_{i+1}(k))}{B_{K}(q_{1}(k), … q_{n}(k))}

where P (x, y) = (x + y)^{2n−2i} − y^{2n−2i}.

We first consider the case that r > i + 1. Then for large k we have that P (q_{i}(k), q_{i+1}(k)) is bounded (in fact eventually constant) so

lim_{k→∞} \frac{B_{K}(ν(x_{k})) − B_{K}(σ_{i}(ν(x_{k})))}{B_{K}(ν(x_{k}))} = lim_{k→∞} \frac{P (q_{i}(k), q_{i+1}(k))}{B_{K}(q_{1}(k), …, q_{n}(k))} = 0

and we have the desired result.

In case r = i + 1 or r = i we observe

\frac{P (q_{i}(k), q_{i+1}(k))}{q_{1}(k)^{2n} + ⋯ + q_{n}(k)^{2} + K} ≤ \frac{P (q_{i}(k), q_{i+1}(k))}{q_{i}(k)^{2n−2i+2} + q_{i+1}(k)^{2n−2i} + K}

and at least one of q_{i}(k)^{2n−2i+2} and q_{i+1}(k)^{2n−2i} tends to infinity so by Lemma 2.2 we again have

lim_{k→∞} \frac{B_{K}(ν(x_{k})) − B_{K}(σ_{i}(ν(x_{k})))}{B_{K}(ν(x_{k}))} = lim_{k→∞} \frac{P (q_{i}(k), q_{i+1}(k))}{B_{K}(q_{1}(k), …, q_{n}(k))} = 0

Finally in case r < i

\frac{|P (q_{i}(k), q_{i+1}(k))|}{B_{K}(q_{1}(k), …, q_{n}(k))} ≤ \frac{|P (q_{i}(k), q_{i+1}(k))|}{q_{r}(k)^{2n−2r+2} + q_{i}(k)^{2n−2i+2} + q_{i+1}(k)^{2n−2i} + K}

and |q_{r}(k)| tends to infinity. But given ε > 0 by Lemma 2.2 there is an M > 0 such that whenever ∥(q_{i}(k), q_{i+1}(k))∥ > M we have

\frac{|P (q_{i}(k), q_{i+1}(k))|}{q_{r}(k)^{2n−2r+2} + q_{i}(k)^{2n−2i+2} + q_{i+1}(k)^{2n−2i} + K}

≤ \frac{|P (q_{i}(k), q_{i+1}(k))|}{q_{i}(k)^{2n−2i+2} + q_{i+1}(k)^{2n−2i} + K} < ε

On the other hand if ∥(q_{i}(k), q_{i+1}(k))∥ ≤ M and |q_{r}(k)| is sufficiently large

\frac{|P (q_{i}(k), q_{i+1}(k))|}{q_{r}(k)^{2n−2r+2} + q_{i}(k)^{2n−2i+2} + q_{i+1}(k)^{2n−2i} + K} < ε

So in all cases we have the desired limit

lim_{k→∞} \frac{|B_{K}(ν(x_{k})) − B_{K}(σ_{i}(ν(x_{k})))|}{B_{K}(ν(x_{k}))} = 0  (4)

which implies

lim_{k→∞} |1 − \frac{B_{K}(σ_{i}(ν(x_{k})))}{B_{K}(ν(x_{k}))}| = 0

and hence that

lim_{k→∞} \frac{B_{K}(σ_{i}(ν(x_{k})))}{B_{K}(ν(x_{k}))} = 1.

⋄

Lemma 2.9. Suppose x_{k} is a monotonic sequence in I_{K} \ J_{K} converging to a point of J_{K}. Then for each 1 ≤ i < n,

lim_{k→∞} g′_{i}(x_{k}) = 1.

Proof. For x ∈ I_{K}(q_{1}, …, q_{n}), g_{i} is defined by g_{i}(x) = ϕ_{b_{1},b_{2}}(x − c_{2}) + c_{1}, for some constants c_{1} and c_{2}, where

b_{1} = b_{1}(k) = \frac{1}{B_{K}(ν(x_{k}))} and b_{2} = b_{2}(k) = \frac{1}{B_{K}(σ_{i}(ν(x_{k})))}

So g′_{i}(x_{k}) = ϕ′_{b_{1},b_{2}}(x_{k}).

We wish to show that given ε > 0 for k sufficiently large |1 − g′_{i}(x_{k})| < ε. To do this, by Lemma 2.1 we need only show that for any δ > 0

|\frac{b_{2}(k)}{b_{1}(k)} − 1| < δ

for sufficiently large k.

But

|\frac{b_{2}(k)}{b_{1}(k)} − 1| = |\frac{B_{K}(ν(x_{k}))}{B_{K}(σ_{i}(ν(x_{k})))} − 1|

which, by Lemma 2.8 tends to 0 as k tends to infinity. Hence

|\frac{b_{2}(k)}{b_{1}(k)} − 1| < δ

for k sufficiently large. ⋄

The proofs of Lemmas (2.10) and (2.11) below are parallel to those of Lemmas (2.8) and (2.9). The difference is that before we were interested in the limiting values of some quantities as a sequence {x_{k}} of values of x in I_{K} \ J_{K} converged, while now we are interested in estimating the same quantities but, uniformly in x, as the parameter K tends to infinity.

Lemma 2.10. Suppose 1 ≤ i < n. Given ε > 0 there is a K_{0} > 0 such that whenever K > K_{0}

|\frac{B_{K}(σ_{i}(ν(x)))}{B_{K}(ν(x))} − 1| < ε

for all x ∈ I_{K} \ J_{K}.

Proof. For 1 ≤ j ≤ n, we define q_{j}(x) by ν(x) = (q_{1}(x), q_{2}(x), …, q_{n}(x)). We note that

B_{K}(σ_{i}(ν(x))) = B_{K}(q_{1}(x), …, q_{i}(x), q_{i+1}(x) + q_{i}(x), q_{i+2}(x), …, q_{n}(x))

and hence if P (u, v) = (u + v)^{2n−2i} − v^{2n−2i}, we have

|\frac{B_{K}(ν(x)) − B_{K}(σ_{i}(ν(x)))}{B_{K}(ν(x)}| = |\frac{(q_{i+1}(x) + q_{i}(x))^{2n−2i} − q_{i+1}(x)^{2n−2i}}{B_{K}(q_{1}(x), … q_{n}(x)} |

= \frac{|P (q_{i}(x), q_{i+1}(x)|}{B_{K}(q_{1}(x), … q_{n}(x))}

≤ \frac{|P (q_{i}(x), q_{i+1}(x))|}{q_{i}(x)^{2n−2i+2} + q_{i+1}(x)^{2n−2i} + K} .

So by the second part of Lemma (2.2) we conclude that if K is sufficiently large

|\frac{B_{K}(ν(x)) − B_{K}(σ_{i}(ν(x)))}{B_{K}(ν(x))}| < ε.  (5)

From this we see

|1 − \frac{B_{K}(σ_{i}(ν(x)))}{B_{K}(ν(x))}| < ε

as desired. ⋄

Lemma 2.11. Given ε > 0, if K is chosen sufficiently large then for all x ∈ I_{K} \ J_{K} and all 1 ≤ i < n

|g′_{i}(x) − 1| < ε.

Proof. For x ∈ I_{K}(q_{1}, …, q_{n}), g_{i} is defined by g_{i}(x) = ϕ_{b_{1},b_{2}}(x − c_{2}) + c_{1}, for some constants c_{1} and c_{2}, where

b_{1} = b_{1}(x) = \frac{1}{B_{K}(ν(x))} and b_{2} = b_{2}(x) = \frac{1}{B_{K}(σ_{i}(ν(x)))}.

So g′_{i}(x) = ϕ′_{b_{1},b_{2}}(x).

We wish to show that given ε > 0, if K is sufficiently large then |1−g′_{i}(x)| < ε. To do this, by Lemma 2.1 we need only show that given any δ > 0 there is a K_{0} > 0 such that K > K_{0} implies

|\frac{b_{2}(x)}{b_{1}(x)} − 1| < δ

for all x ∈ I_{K} \ J_{K}.

But

|\frac{b_{2}(x)}{b_{1}(x)} − 1| = |\frac{B_{K}(σ_{i}(ν(x)))}{B_{K}(ν(x))} − 1|

and the result follows from Lemma 2.10. ⋄

Proposition 2.12. For K sufficiently large the homeomorphism g_{i} : I_{K} → I_{K} is a C^{1} diffeomorphism with derivative 1 at both endpoints. Given ε > 0 there exists K_{0} such that whenever K > K_{0} we have

|g′_{i}(x) − 1| < ε

for all x ∈ I_{K}.

Proof. We know that the function f(x) = g′_{i}(x) exists and is continuous on I_{K} \ J_{K}. By Lemma (2.9) we can extend it continuously to all of I_{K} by setting f(x) = 1 for x ∈ J_{K}.

We define a C^{1} function F by

F (x) = ∫_{0}^{x} f(t)dt

and will show that F (x) = g_{i}(x). To see this let ϕ(x) = g_{i}(x) − F (x). Then ϕ(0) = 0 and ϕ(x) is a continuous function whose derivative exists and is 0 on I_{K} \ J_{K}. Since J_{K} is countable ϕ(J_{K}) has measure zero. But I_{K} \ J_{K} has countably many components on each of which ϕ is constant. It follows that ϕ(I_{K}) has measure zero and hence ϕ(I_{K}) = {0}. Therefore g_{i}(x) = F (x) is C^{1}.

Lemma (2.11) and the fact that g′_{i}(x) = 1 for x ∈ J_{K} imply that for K sufficiently large

|g′_{i}(x) − 1| < ε

for all x ∈ I_{K}. The inverse function theorem then implies that g_{i} is a C^{1} diffeomorphism. ⋄

Recall that we have given the group Z^{n} the lexicographic ordering, ≻ i.e. (x_{1}, …, x_{n}) ≻ (y_{1}, …, y_{n}) if and only if x_{i} = y_{i} for 1 ≤ i < k and x_{k} > y_{k} for some 0 ≤ k ≤ n. We note that this order is translation invariant, indeed (x_{1}, …, x_{n}) ≻ (y_{1}, …, y_{n}) if and only if (x_{1} − y_{1}, …, x_{n} − y_{n}) ≻ (0, …, 0).

It is well known and easy to check that the nilpotent group N_{n} of n×n lowertriangular integer matrices with ones on the diagonal acts on Z^{n} preserving the order ≻.

Theorem 2.13. The group N_{n} is isomorphic to a subgroup of Diff^{1}(M) for M = R, S^{1}, or I. For M = S^{1} or I the elements of this subgroup corresponding to the generators {σ_{i}} of N_{n} may be chosen to be in an arbitrary neighborhood of the identity in Diff^{1}(M).

Proof. We first consider the case that M = I. Given ε > 0 we choose K sufficiently large that the conclusion of Proposition (2.12) holds. We define Φ : N_{n} → Homeo(I_{K}) by Φ(α) = g_{α}. Lemma (2.6) asserts that Φ is an injective homomorphism. Proposition (2.12) asserts that g_{i} = Φ(σ_{1}) is a C^{1} diffeomorphism so in fact Φ(N_{n}) lies in Diff^{1}(I_{K}).

Define the injective homomorphism Ψ : N_{n} → Diff^{1}(I) by Ψ(α)(x) = S_{K}Φ(α)(x/S_{K}) = S_{K}g_{α}(x/S_{K}). Restricting this action to the interior of I givens an action on R.

Note that every element of Ψ(N_{n}) has derivative 1 at both endpoints of I. Gluing endpoints together gives an action on S^{1}.

In the case M = S^{1} or I we clearly have |Ψ(σ_{i})′(x) − 1| < ε by Proposition (2.12) ⋄

Now every finitely-generated, torsion-free nilpotent group N is isomorphic to a subgroup of N_{n} for some n (see Theorem 4.12 of [Ra] and its proof). This together with Theorem 2.13 immediately implies Theorem 1.2.

2.3 C^{∞} actions on R

For certain nilpotent groups we can give a C^{∞} action on R. But we will see in the next section that for n > 2 there is no C^{2} action of N_{n} on R.

Proof of Theorem 1.3: Choose a non-trivial C^{∞} diffeomorphism α of [0, 1] to itself such that for j = 0, 1 we have α(j) = j, α′(j) = 1, and α^{[k]}(j) = 0, for all k > 1.

Define three C^{∞} diffeomorphisms f, h_{0}, h_{1} : R → R by

f(x) = x − 1
h_{0}(x) = α(x − m) + m  for x ∈ [m, m + 1]
h_{1}(x) = α^{m}(x − m) + m for x ∈ [m, m + 1]

Then it is easy to check that h_{0} and h_{1} commute, f and h_{0} commute, and [f, h_{1}] = f^{−1}h_{1}^{−1}fh_{1} = h_{0}. Hence the group generated by f and h_{1} is nilpotent with degree of nilpotency 2.

Given any n > 0, we will inductively define h_{k} for 0 ≤ k ≤ n in such a way that h(m) = m for m ∈ Z, [f, h_{k}] = h_{k−1} for k > 1, and h_{i}h_{j} = h_{j}h_{i}. We do this by letting h_{k}(x) = x for x ∈ [0, 1] and recursively defining h_{k}^{−1}. We first define it for x > 1 by

h_{k}^{−1}(x) = h_{k−1}^{−1}f^{−1}h_{k}^{−1}f(x) for x > 1.

Note this is well defined recursively because if x ∈ [n, n + 1] the right hand side requires only that we know the value of h_{k}^{−1}(f(x)) and f(x) ∈ [n − 1, n]. We also note that h_{k}^{−1} = h_{k−1}^{−1}f^{−1}h_{k}^{−1}f implies h_{k−1}h_{k}^{−1} = f^{−1}h_{k}^{−1}f, so h_{k−1}(x) = f^{−1}h_{k}^{−1}fh_{k}(x) for all x > 1.

Negative values of x are handled similarly. We define

h_{k}^{−1}(x) = fh_{k−1}h_{k}^{−1}f^{−1}(x) for x < 0.

This is well defined recursively for x < 0 because if x ∈ [−n, −n + 1] the right hand side requires only that we know the value of h_{k}^{−1}(f^{−1}(x)) and f^{−1}(x) ∈ [−n + 1, −n + 2]. Again h_{k}^{−1} = fh_{k−1}h_{k}^{−1}f^{−1} implies f^{−1}h_{k}^{−1}f = h_{k−1}h_{k}^{−1} so h_{k−1}(x) = f^{−1}h_{k}^{−1}fh_{k}(x) for all x < 0.

We further note that from this definition one sees easily inductively that h_{k}(x) = x for x ∈ [−1, 0] for all k ≥ 2. Finally we observe that this implies for x ∈ [0, 1] we have f^{−1}h_{k}^{−1}fh_{k}(x) = x so f^{−1}h_{k}^{−1}fh_{k}(x) = h_{k−1}(x).

It is clear from the relations [f, h_{k}] = h_{k−1} for k > 1, [f, h_{0}] = id, and h_{i}h_{j} = h_{j}h_{i} that the group generated by f and the h_{k} is nilpotent of degree at most n. The degree of nilpotency is at least n since h_{0} is a nontrivial n-fold commutator. ⋄

If n = 2, the group G constructed above is isomorphic to the group N_{3} of 3×3 lower triangular matrices. In general, G is isomorphic to a semi-direct product of Z^{n} and Z where the Z action on Z^{n} is given by the n × n lower triangular matrix with ones on the diagonal and subdiagonal and zeroes elsewhere. It is metabelian, i.e. solvable with derived length two. We will see below that this is a necessary condition for a smooth action on R.

2.4 Residually torsion-free nilpotent groups

In this section we consider some actions on R which are not irreducible, but which are actions of a fairly large class groups.

Definition 2.14. A group G is called residually torsion-free nilpotent if, for every non-trivial element g ∈ G, there is a torsion-free nilpotent group N and a homomorphism ϕ : G → N such that ϕ(g) is non-trivial. Equivalently, the intersection of every term in the lower central series for G is trivial.

Proof of Theorem 1.6: Since G is finitely generated it contains countably many elements. Let {g_{m} | i ∈ Z^{+}} be an enumeration of the non-trivial elements of G. Let ϕ_{m} : G → N_{m} be the homomorphism to a torsion free nilpotent group guaranteed by residual nilpotence, so ϕ_{m}(g_{m}) is non-trivial. Replacing N_{m} by ϕ_{m}(G) if necessary we may assume N_{m} is finitely generated.

Let I_{m} = [1/(m + 1), 1/m]. Using Theorem 1.2, choose an effective action of N_{m} by C^{1} diffeomorphisms on the interval I_{m}. Recall that this action has the property that the derivative of every element is 1 at the endpoints of I_{m}. In addition, by Theorem 1.2, we may choose this action so that the derivative of ϕ_{m}(g_{i}) satisfies

|ϕ_{m}(g_{i})′(x) − 1| < \frac{1}{2^{m}}

for all x ∈ I_{m} and all 1 ≤ i ≤ m.

We then define the action of G by g(x) = ϕ_{m}(g)(x) for x ∈ I_{m} and g(0) = 0 for all g ∈ G. Clearly each g_{m} is a C^{1} diffeomorphism with g′_{m}(0) = 1. The action is effective because ϕ_{m}(g) acts non-trivially on I_{m}.

Since every element of G has derivative 1 at both endpoints of I, we may glue the endpoints together to give a C^{1} action on S^{1}. ⋄

It is an old result of Magnus that free groups and surface groups are residually torsion-free nilpotent. In particular Theorem 1.6 gives a C^{1} action of surface groups on R, I and S^{1}; we do not know another proof that such actions exist.

Another interesting example is the Torelli group T_{g}, which is defined to be the kernel of the natural action of the mapping class group of a genus g surface Σ_{g} on H_{1}(Σ_{g}, Z). For g ≥ 3, D. Johnson proved that T_{g} is finitely-generated, and Bass-Lubotzky proved that T_{g} is residually torsion-free nilpotent. Similarly, the kernel of the action of the outer automorphism group of a free group is finitely-generated and residually torsion-free nilpotent. Hence by Theorem 1.6 both of these groups are subgroups of Diff^{1}(M) for M = R, I, S^{1}. In particular the Torelli group is left orderable. It is not known wether or not mapping class groups are left-orederable, althought Thurston has proven that braid groups are left-orderable.

3 Restrictions on C^{2} actions

In the previous sections we showed that actions by nilpotent subgroups of homeomorphisms are abundant. By way of contrast in the next two sections we show that nilpotent groups of C^{2} diffeomorphisms or PL homeomorphisms are very restricted. In this section we focus on C^{2} actions.

3.1 Kopell’s Lemma

Our primary tool is the following remarkable result of Nancy Kopell, which is Lemma 1 of [K].

Theorem 3.1 (Kopell’s Lemma). Suppose f and g are C^{2} orientation preserving diffeomorphisms of an interval [a, b) or (a, b] and fg = gf. If f has no fixed point in (a, b) and g has a fixed point in (a, b) then g = id.

We will primarily use a consequence of this result which we now present.

Definition 3.2. We will denote by ∂ Fix(f) the frontier of Fix(f), i.e. the set ∂ Fix(f) = Fix(f) \ Int(Fix(f)).

Lemma 3.3. Suppose f and g are commuting orientation preserving C^{2} diffeomorphisms of R, each of which has a fixed point. Then f preserves each component of Fix(g) and vice versa. Moreover, ∂ Fix(g) ⊂ Fix(f) and vice versa. The same result is true for C^{2} diffeomorphisms of a closed or half-open interval, in which case the requirement that f and g have fixed points is automatically satisfied by an endpoint.

Proof. The proof is by contradiction. Assume X is a component of Fix(g) and f(X) ≠ X. Since f and g commute f(Fix(g)) = Fix(g) so f(X) is some component of Fix(g). Hence f(X) ≠ X implies f(X) ∩ X = ∅. Let x be an element of X and without loss of generality assume f(x) < x. Define

a = lim_{n→∞} f^{n}(x) and b = lim_{n→−∞} f^{n}(x).

Then at least one of the points a and b is finite since f has a fixed point. Also a and b (if finite) are fixed under both f and g, and f has no fixed points in (a, b). Thus Kopell’s Lemma 3.1 implies g(y) = y for all y ∈ (a, b) contradicting the hypothesis that X is a component of Fix(g). The observation that ∂ Fix(f) ⊂ Fix(g) follows from the fact that components of Fix(f) are either points or closed intervals so x ∈ ∂ Fix(f) implies that either {x} is a component of Fix(f) or x is the endpoint of an interval which is a component of Fix(f) so, in either case, the fact that g preserves this component implies x ∈ Fix(g).

The proof in the cases that f and g are diffeomorphisms of a closed or half-open interval is the same. ⋄

Another useful tool is the following folklore theorem (see, e.g., [FS] for a proof).

Theorem 3.4 (Hölder’s Theorem). Let G be a group acting freely and effectively by homeomorphisms on any closed subset of R. Then G is abelian.

3.2 Measure and the translation number

A basic property of finitely generated nilpotent groups of homeomorphisms of a one-manifold M is that they have invariant Borel measures. Of course in the case M = I or T^{1} this follows from the fact that nilpotent groups are amenable and any amenable group acting on a compact Hausdorf space has an invariant Borel probability measure. In the case M = R this is a special case of a result due to J. Plante [P] who showed that any finitely generated subgroup of Homeo(R) with polynomial growth has an invariant measure which is finite on compact sets.

We summarize these facts in the following

Theorem 3.1. Let M = R, S^{1}, or I. Then any finitely generated nilpotent subgroup of Homeo(M) has an invariant Borel measure μ which is finite on compact sets.

If one has an invariant measure for a subgroup of Homeo(R), it is useful to consider the translation number which was discussed by J. Plante in [P]. It is an analog of the rotation number for circle homeomorphisms.

Suppose G is a subgroup of Homeo(R) which preserves a Borel measure μ that is finite on compact sets. Fix a point x ∈ R and for each f ∈ G define

τ(f) = \begin{cases}μ([x, f(x)) & if x < f(x) \\ 0 & if x = f(x) \\ −μ([f(x), x)) & if x > f(x) \end{cases}

The function τ : G → R is called the translation number. The following properties observed by J. Plante in [P] are easy to verify.

Proposition 3.5. The translation number τ : G → R is independent of the choice of x ∈ R used in its definition. It is a homomorphism from G to the additive group R. For any f ∈ G the set Fix(f) ≠ ∅ if and only if τ(f) = 0.

3.3 Proof of Theorem 1.1

Theorem 1.1 Any nilpotent subgroup of Diff^{2}(I), Diff^{2}([0, 1)) or Diff^{2}(S^{1}) must be abelian.

Proof. Since it suffices to show the action is abelian when restricted to any invariant interval we may assume our action is irreducible.

3.3.1 The case M = I or M = [0, 1)

The argument we present here is essentially the same as that given by Plante and Thurston in (4.5) of [PT]. We give it here for completeness and because it is quite simple.

Consider the restriction of N to (0, 1). If no element of N has a fixed point then N is abelian by Hölder’s theorem. Hence we may assume that there is a non-trivial element f with a fixed point. Thus, if h is in the center of N Lemma 3.3 implies that Fix(h) ⊃ ∂ Fix(f) which is non-empty. Another appliction of Lemma 3.3, says the non-empty set ∂ Fix(h) ⊂ Fix(g) for any g ∈ N. We have found a global fixed point and contradicted the assumption that N acted irreducibly. We conclude N is abelian. ⋄

3.3.2 The case M = S^{1}

Let N < Diff^{2}(S^{1}) be nilpotent. As we observed in Theorem 3.1 there is an invariant measure μ_{0} for the action. If any one element has no periodic points then since it is C^{2} it is topologically conjugate to an irrational rotation by Denjoy’s theorem (see e.g. [dM]). Irrational rotations are uniquely ergodic so the invariant measure must be conjugate to Lebesgue measure. It follows that every element is conjugate to a rotation so N is abelian. Hence we may assume that every element of N has periodic points.

For a homeomorphism of the circle with periodic points, every point which is not periodic is wandering and hence not in the support of any invariant measure. We conclude that the periodic points of any element contain the support P of the measure μ_{0}. It follows that that P is a subset of the periodic points of every element of N.

Since N preserves μ_{0} the group N̂ of all lifts to R of elements of N preserves a measure μ which is the lift of the measure μ_{0} on S^{1}. We can use μ to define the translation number homomorphism τ_{μ}N̂ → R as described above.

We observe that commutators of elements in N̂ must have translation number 0. The covering translations in N̂ are in its center so if f̂ and ĝ are lifts of f and g respectively then [f̂, ĝ] is a lift of [f, g]. Since τ_{μ}([f̂, ĝ]) = 0 implies [f̂, ĝ] has a fixed point we may conclude that every element of the commutator subgroup N_{1} = [N, N] must have a fixed point. Hence any commutator fixes every element of P because it fixes one point and hence all its periodic points are fixed.

If N is not abelian there are f and g in N such that h = [f, g] is a non-trivial element of the center of N. Conjugating the equation f^{−1}g^{−1}fg = h by f gives g^{−1}fgf^{−1} = h, so fgf^{−1} = hg. Repeatedly conjugating by f gives f^{n}gf^{−n} = h^{n}g. From this we get g^{−1}f^{n}g = h^{n}f^{n} and by repeatedly conjugating with g we obtain g^{−m}f^{n}g^{m} = h^{mn}f^{n}. We conclude g^{−m}f^{n}g^{m}f^{−n} = [g^{m}, f^{−n}] = h^{mn}. Now let x ∈ P and let m and n be its period under the maps f and g respectively. Then h, f^{n} and g^{m} all fix the point x. If we split the circle at x we get an interval and a C^{2} nilpotent group of diffeomorphisms generated by h, f^{n} and g^{m}. By Theorem 1.1 this group is abelian. Hence [g^{m}, f^{−n}] = h^{mn} is the identity. But the only finite order orientation-preservng homeomorphism of an inteval is the identity. We have contrdicted the assumption that N is not abelian. ⋄

3.4 The proof of Theorem 1.5

Theorem 1.5 If N is a nilpotent subgroup of Diff^{2}(R) and every element of N has a fixed point then N is abelian.

Proof. We first show that there is a global fixed point for N. Consider Fix(h) the fixed point set of some non-trivial element h of the center of N. We can apply Lemma 3.3, and observe that for any f ∈ N, ∂ Fix(h) ⊂ Fix(f). Thus the non-empty set ∂ Fix(h) is fixed pointwise by every element of N.

If a ∈ Fix(h) then both [a, ∞) and (−∞, a] are invariant by N and each is diffeomorphic to [0, 1). We can thus apply Theorem 1.1 and conclude the restriction of N to each of them is abelian. ⋄

3.5 The proof of Theorem 1.4

Theorem 1.4 Every nilpotent subgroup of Diff^{2}(R) is metabelian, i.e. has abelian commutator subgroup.

Proof. Let N < Diff^{2}(R) be nilpotent. By Theorem 3.1 there is an invariant Borel measure μ on R which is finite on compact sets and from it we can define a translation number homomorphism τ_{μ} : N → R. According to Proposition 3.5 If N_{0} is the kernel of τ_{μ} then every element of N_{0} has a fixed point. Hence the result follows from Theorem 1.5. ⋄

4 PL actions

Recall that a piecewise linear homeomorphism of a one-manifold M is a homeomorphism f for which there exist finitely many subintervals of M on which f is linear. Let PL(M) denote the group of piecewise-linear homeomorphisms of M.

Theorem 4.1. Any nilpotent subgroup of PL(I) or PL(S^{1}) is abelian.

Proof. Suppose first that N < PL(I). Note that there is a natural homomorphism ψ : PL(I) → R^{∗} × R^{∗} given by

ψ(f) = (f′(0), f′(1))

As R × R is abelian ψ(f) = (1, 1) for any nontrivial commutator in PL(I).

If N is not abelian then there is a nontrivial commutator f lying in the center of N. Hence f′(0) = 1, so that f is the identity on some interval [0, a] which we take to be maximal. Since every element of N commutes with f, we must have that g([0, a]) = [0, a] for every g ∈ N. But then a is also a fixed point of every element of N, so that f is a nontrivial commutator of elements which fix a. We conclude that f′(a) = 1 also. Hence f is the identity on [a, b] for some b > a and we contradict the maximality of a. We conclude that the center of N has no nontrivial commutator, so N is abelian.

Now if N < PL(S^{1}) then the proof is nearly word for word identical to the proof of Theorem 1.1. We note that Denjoy’s Theorem (see [dM]) is also valid for PL homeomorphisms of the circle. That is, any PL homeomorphism of S^{1} without periodic points is topologically conjugate to an irrational rotation. With that observation the proof that a PL action of N on S^{1} is abelian is identical to the proof of Theorem 1.1 except that the appeal to Theorem 1.1 in the case of the interval is replaced by an appeal to the first part of this theorem, namely the fact that a PL action of a nilpotent group on I is abelian. ⋄

References

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[FF2] B. Farb and J. Franks, Group actions on one-manifolds, II: Extensions of Hölder’s Theorem, preprint, April 2001.

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[P] J. Plante, Solvable Groups acting on the line, Trans. Amer. Math. Soc. 278, (1983) 401–414.

[PT] J. Plante and W. Thurston, Polynomial Growth in Holonomy Groups of Foliations Comment. Math. Helvetica 39 51 567-584.

[dM] Welington de Melo, Lectures on One-dimensional Dynamics Intstituto de Matematica Pura e Aplicada do CNPq, Rio de Janeiro, (1989) p.137-178.

[MR] R. Botto Mura and A. Rhemtulla, Orderable groups, Marcel Dekker, 1977.

[Ra] M.S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik, Band 68. Springer-Verlag, 1972.

[Wi] D. Witte, Arithmetic groups of higher Q-rank cannot act on 1manifolds, Proc. AMS, Vol.122, No.2, Oct. 1994, pp.333-340.

Benson Farb: Dept. of Mathematics, University of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: farb@math.uchicago.edu

John Franks: Dept. of Mathematics, Northwestern University Evanston, IL 60208 E-mail: john@math.northwestern.edu