Luân án tiến sĩ rigidity of solvable group actions

NORTHWESTERN UNIVERSITY Rigidity of Solvable Group Actions A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Mathematics By Anne E. McCarthy EVANSTON, ILLINOIS June 2006 UMI Number: 3212800 Copyright 2006 by McCarthy, Anne E. All rights reserved. UMI Microform 3212800 Copyright 2006 by ProQuest Information and Learning Company.

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McCarthy 2006 All Rights Reserved 3 ABSTRACT Rigidity of Solvable Group Actions Anne E. McCarthy This thesis investigates dynamical properties of actions of abelian-by-cyclic groups on compact manifolds. For a non-singular integer matrix A, let ΓA be the fundamental group of the mapping cylinder of the induced map fA on the torus T n. The standard actions ρλ of ΓA on the circle RP 1 are generated by maps f (x) = λx and gi (x) = x + bi , where λ is a real-valued eigenvalue for A, and (β1 , ., βn ) is the associated eigenvector.

It is known that any analytic action of ΓA on the circle is a ramified lift of one of the standard actions ρλ. This thesis shows that for each analytic action, ρ, there exists R ≥ 2 such that ρ is C r locally rigid for all r ≥ R. We then consider actions of the groups ΓA on compact manifolds of higher dimension that are generated by C 1 diffeomorphisms close to the identity. We show that any action of ΓA on a surface with non-zero Euler characteristic has a global fixed point.

Also, we show that for any compact manifold M , there are no faithful actions of the Baumslag-Solitar group generated by diffeomorphisms close to the identity. 4 Acknowledgements Many thanks are due to my advisor, Amie Wilkinson. Under her guidance and in- struction, I have been fortunate to have absorbed even a small fraction of her breadth and depth in knowledge of mathematics. I also am grateful to have been the sometimes undeserving beneficiary of both her paitence and generosity.

Benson Farb posed many of the questions that have inspired the work of this thesis. I extend gratitude to Benson for many illuminating conversation about Sol and BS groups. Keith Burns has enriched my study of dynamics at Northwestern by overseeing many working seminars in which I have been priveledged to participate. Several conversations with John Franks have also been very helpful.

I would also like to thank Michael Johnson, Kalman Nanes, Chris Novak, and all the other graduate students who have taken the time to listen to me talk about my work, and have given useful feedback on many of the talks I have given in the past year. 5 Table of Contents ABSTRACT 3 Acknowledgements 4 Chapter 1. Abelian-by-cyclic actions on S 1 15 2. Global Invariant Sets 21 2.

The Liouville cocycle and BS(1,n) 38 2. Ramified Lifts of Affine Actions 41 2. Proof of Theorem 1. Actions Not Satisfying the Spectral Gap Condition 51 Chapter 3.

Actions of Abelian-by-Cyclic Groups on Manifolds 53 3. Invariant Sets for Actions of ΓA Close to the Identity 55 3. Rigidity for BS(1, n) close to id 57 3. Actions of Subgroups of Aff(R) on S 2 59 6 3.

Examples of BS(1, n) acting on S 2 60 References 65 7 CHAPTER 1 Introduction This thesis investigates the dynamics of solvable group actions on compact manifolds. The actions of nilpotent and solvable groups on one-manifolds have been studied by Kopell [K], Plante and Thurston [PT], Ghys [G], and Farb and Franks [FF1]. Furthermore, work of Navas [N1] and Burslem and Wilkinson [BW] gives a classification for solvable group actions on S 1 under the appropriate regularity hypotheses. We add to this work by demonstrating a class of solvable groups that exhibit unusual rigidity phenomena.

We then turn to the study of actions of solvable groups on higher dimensional manifolds. This thesis includes an investigation of actions of abelian-by-cyclic groups on any manifold that are generated by diffeomorphisms close to the identity. Given a finitely generated group Γ and a manifold M , a C r action of Γ on M is a homomorphism ρ : Γ → Diff r (M ). We will commonly refer to this homomorphism as a representation (into Diff r (M )) and use the associated language.

The representation ρ is said to be faithful if ρ is injective. We denote by Rr (Γ, M ) the collection of all representations of Γ into Diff r (M ). A point q ∈ M is said to be a global fixed point for the action ρ if q is fixed by ρ(γ) for all γ ∈ Γ. The goal of this thesis is to study the structure of Rr (Γ, M ) where Γ is a solvable group, and M is a compact manifold.

The central motivating examples of this thesis are 8 actions on the circle S 1 of the solvable Baumslag-Solitar group: BS(1, n) = ha, b|aba−1 = bn i. The standard affine action of this group on the circle S 1 ' RP 1 given by ρ(a) : x 7→ nx and ρ(b) : x 7→ x + 1. This standard action can be used to generate many more actions by taking ramified lifts. f → M is said to be a ramified covering map over the point An analytic map π : M f\{π −1 (p)} is a regular covering map onto M \{p}.

A p ∈ M , if the restriction of π to M f→M map fb : M f is said to be a π-ramified lift of f : M → M if π ◦ fb = f ◦ π for the ramified covering π. For any ramified covering map π : S 1 → S 1 there exists a π-ramified lift ρbn : BS(1, n) → Diff ω (S 1 ) of the standard representation, ρn. It was shown by Burslem and Wilkinson [BW] that any faithful action ρ : BS(1, n) → Diff ω (S 1 ) is conjugate to a π-ramified lift of the standard action ρn. Furthermore, [BW] also prove that all analytic actions ρ : BS(1, n) → Diff ω are locally rigid, as defined below.

The collection of C r representations, Rr (Γ, M ) carries a topology. For the group Γ, fix a generating set hγ1 , ., γk i, and let dC r (ρ1 , ρ2 ) = sup dC r (ρ1 (γi ), ρ2 (γi )).,γk This topology is independent of the choice of generating set. Representations ρ, ρ0 ∈ Rr (Γ, M ) are said to be conjugate in Diff r (M ) if there exists h ∈ Diff r (M ) such that h ◦ ρ(γ) = ρ0 (γ) ◦ h for all γ ∈ Γ. A representation ρ0 ∈ Rr (Γ, M ) is C (k,r) locally rigid 9 if there exists a C k -open neighborhood U of ρ0 in Rr (Γ, M ) such that every ρ ∈ U is conjugate in Diff r (M ) to ρ0.

The authors [BW] also give a classification of all analytic actions on S 1 of solvable groups. Suppose that G < Diff ω (S 1 ) is solvable. Then either G is virtually abelian, or G is conjugate in Diff ω (S 1 ) to a subgroup of a π-ramified lift of Aff(R), where π : RP 1 → RP 1 is a ramified cover over ∞. This thesis shows that many of the results of [BW] about actions of BS(1, n) on S 1 extend to the class of abelian-by-cyclic groups.

To any invertible n × n matrix A with integer entries, one can associate the solvable group Y a ΓA = ha, b1 , .bn |bi bj = bj bi , abi a−1 = bj ij i. j This group has a geometric interpretation as the fundamental group of the mapping cylinder for the map induced by A on Tn. Note that in the case where A is the 1 × 1 matrix A = [n], we have that ΓA = BS(1, n). A group Γ is said to be abelian-by-cyclic if there exists an exact sequence 1 → A → Γ → Z → 1, where the group A is abelian, and Z is an infinite cyclic group.

Note that the commutator subgroup [Γ, Γ] is contained in A, so all such Γ are solvable groups. The class of all finitely 10 presented, torsion free, abelian-by-cyclic groups is exactly given by groups of the form ΓA. See [FM] for a nice proof of this. To each real eigenvalue of A, there corresponds an action of ΓA on the circle, given by ρλ (a) : x 7→ λx and ρλ (bj ) : x 7→ x + βj , where βj are the entries of the eigenvector corresponding to the eigenvalue λ, normalized in such a way that β1 = 1.

We call the representations ρλ the standard representations of ΓA into Aff(R) < Diff ω (S 1 ). The classification of real-analytic actions of solvable groups given in [BW] implies that each action ρ : ΓA → Diff ω (S 1 ) is conjugate to a π-ramified lift of one of these standard affine actions. Our first main result is a classification of representations ρ : ΓA → Diff r (S 1 ) for r < ω. The main hypothesis of our result involves the hyperbolicity of the representation in the following sense.

We require that ρ have a spectral gap compatible with the spectrum of the matrix A, as follows. Let the eigenvalues of A be given as |λn | ≥. ≥ |λ1 |, and fix generators ha, bi i for the group ΓA. We will say that a representation ρ ∈ Rr (S 1 ) satisfies the C r spectral gap condition if 1 1 σ − (ρ) < ( |λ1n | ) r−1 and σ + (ρ) > ( |λ11 | ) r−1.

With this we now can state our main theorem concerning C r actions of abelian-by-cyclic groups on the circle. Suppose that A is a non-singular integer matrix with no eigenvalues that are roots of unity. For each r ≥ 2, if ρ : ΓA → Diff r (S 1 ) is a faithful representation 11 that satisfies the C r spectral gap condition, then ρ is conjugated by an element of Diff r (S 1 ) into a unique conjugacy class of Rω (ΓA ). The C r spectral gap condition serves to prohibit the group Γ from acting via diffeo- morphisms that are r-tangent to the identity.

The classification given by Theorem 1.2 does not apply to actions with diffeomorphisms tangent to the identity. Furthermore, in many cases, including that of BS(1, n) actions that do not satisfy the C r spectral gap condition are not C (1,r) locally rigid. One feature that is common to all faithful actions of the groups ΓA on the circle is the existence of a finite globally invariant set. This and other features enjoyed by circle actions no longer remain true if we instead consider actions of BS(1, n) on higher dimensional manifolds.

The question of existence of global fixed points for group actions on surfaces has been studied in many different contexts. The earliest work in this area concerned global fixed points for actions of Lie groups on closed surfaces with non-zero Euler characteristic. It was shown by Lima [L] that n commuting vector fields on a surface Σg of non-zero Euler characteristic have a common singularity. This implies that any action of the abelian Lie group Rn on Σg has a global fixed point.