Nonsmoothable, locally indicable group actions on the interval
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.
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 ).
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 ).
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.
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 , or when there is no invariant probability measure  (also, see ).
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.
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  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.
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 .
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
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. ∎
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.
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. .
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