# Nonsmoothable, locally indicable group actions on the interval

###### Abstract.

By the Thurston stability theorem, a group of orientation-preserving diffeomorphisms of the closed unit interval is locally indicable. We show that the local order structure of orbits gives a stronger criterion for nonsmoothability that can be used to produce new examples of locally indicable groups of homeomorphisms of the interval that are not conjugate to groups of diffeomorphisms.

## 1. Introduction

### 1.1. Acknowledgment

This note was inspired by a comment in a lecture by Andrés Navas. I would like to thank Andrés for his encouragement to write it up. I would also like to thank the referee, whose many excellent comments have been incorporated into this paper.

## 2. Nonsmoothable actions

### 2.1. Thurston stability theorem

A simple, but important case of the Thurston Stability Theorem is usually stated in the following way:

###### Theorem 2.1 (Thurston Stability Theorem [8]).

Let be a group of orientation-preserving diffeomorphisms of the closed interval . Then is locally indicable; i.e. every nontrivial finitely generated subgroup of admits a surjective homomorphism to .

The proof is non-constructive, and uses the axiom of choice. The idea is to “blow up” the action of near one of the endpoints at a sequence of points that are moved a definite distance, but not too far. Some subsequence of blow-ups converges to an action by translations.

Note that it is only finitely generated subgroups that admit surjective homomorphisms to , as the following example of Sergeraert shows.

###### Example 2.2 (Sergeraert [7]).

Let be the group of orientation-preserving diffeomorphisms of that are infinitely tangent to the identity at the endpoints. Then is perfect.

Another countable example comes from Thompson’s group.

###### Example 2.3 (Navas [6], Ghys-Sergiescu [3]).

Thompson’s group of dyadic rational piecewise linear homeomorphisms of is known to be conjugate to a group of diffeomorphisms. On the other hand, the commutator subgroup is simple; since it is non-Abelian, it is perfect.

Given a group , Theorem 2.1 gives a criterion to show that the action of is not conjugate into . It is natural to ask whether Thurston’s criterion is sharp. That is, suppose is locally indicable. Is it true that every homomorphism from into is conjugate into ? It turns out that the answer to this question is no. However, apart from Thurston’s criterion, very few obstructions to conjugating a subgroup of into are known. Most significant are dynamical obstructions concerning the existence of elements with hyperbolic fixed points when the action has positive topological entropy [4], or when there is no invariant probability measure [2] (also, see [1]).

In this note we give some new examples of actions of locally indicable groups on that are not conjugate to actions.

###### Example 2.4 ().

Let act freely on the interior, so that is conjugate to a translation. Let be a closed fundamental domain for , and let act freely on the interior. Extend by the identity outside to an element of . For each let and let be the conjugate . For each define to be the product

Let be the group consisting of all elements of the form . Then is isomorphic to and is therefore abelian.

However, is not conjugate into . For, suppose otherwise, so that there is some homeomorphism so that the conjugate . We suppose by abuse of notation that denotes the conjugate . For each , let be the midpoint of . Since for each fixed the sequence converges to an endpoint of as goes to infinity, it follows that for each there is some so that . Let satisfy . Then for all . However, fixes the endpoints of for all , so has a sequence of fixed points converging to . It follows that . But , so if is we must have . This contradiction shows that no such conjugacy exists.

###### Remark 2.5.

The group is locally indicable, but uncountable. Note in fact that this group action is not even conjugate to a bi-Lipschitz action. On the other hand, Theorem D from [2] says that every countable group of homeomorphisms of the circle or interval is conjugate to a group of bi-Lipschitz homeomorphisms.

### 2.2. Order structure of orbits

In this section we describe a new criterion for non-smoothability, depending on the local order structure of orbits.

###### Definition 2.6.

Let act on by . A point determines an order on by

if and only if in .

Note that with this definition, is really an order on the left -space , where denotes the stabilizer of .

###### Lemma 2.7.

Suppose is injective. Let be a finitely generated subgroup of , with generators . Let be in the frontier of (i.e. the set of common fixed points of all elements of ) and let be a sequence contained in . Then there is a sequence and such that for any , and for all sufficiently large (depending on ), there is an inequality

###### Proof.

There is a homomorphism defined by the formula . Of course this homomorphism vanishes on . If is such that then (after replacing by if necessary) it is clear that for any , there is an inequality for all sufficiently close to . Therefore in the sequel we assume is trivial.

For each , let be the smallest (closed) interval containing . Given a bigger open interval containing , one can rescale linearly by and move to the origin thereby obtaining an interval on which has a partially defined action as a pseudogroup.

The argument of the Thurston stability theorem implies that one can choose a sequence such that any sequence of indices contains a subsequence for which , and the pseudogroup actions converge, in the compact-open topology, to a (nontrivial) action of on by translations. In an action by translations, some generator or its inverse moves a positive distance, but every element of acts trivially. The proof follows. ∎

###### Example 2.8.

Let be a hyperbolic once-punctured torus with a cusp. The hyperbolic structure determines up to conjugacy a faithful homomorphism .

The group acts by real analytic homeomorphisms on . Since is free on two generators (say ) the homomorphism lifts to an action on the universal cover . We choose a lift so that both and have fixed points. If we choose co-ordinates on so that fixes , then also fixes for every integer . Similarly, if fixes , then fixes for every . On the other hand, if is the parabolic fixed point of , and is a lift of to , then the commutator takes to . Since the action of every element on commutes with the generator of the deck group , the element acts on without fixed points, and moves every point in the positive direction, satisfying for every and every positive integer . See Figure 1.

This action on can be made into an action on by homeomorphisms, by including in as the interior. Then the points in map to points in . Note that for each , the elements and have fixed points respectively satisfying and . Moreover, for all . It follows that

for every , so by Lemma 2.7, this action is not topologically conjugate into . On the other hand, this is a faithful action of the free group on two generators. A free group is locally indicable, since every subgroup of a free group is free.

###### Remark 2.9.

The relationship between order structures and dynamics of subgroups of homeomorphisms of the interval is subtle and deep. For an introduction to this subject, see e.g. [5].

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