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 3212800 2006 Copyright 2006 by McCarthy, Anne E. UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 All rights reserved. by ProQuest Information and Learning Company. 2 c  Copyright by Anne E. 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 Γ A be the fundamental group of the mapping cylinder of the induced map f A on the torus T n . The standard actions ρ λ of Γ A on the circle RP 1 are generated by maps f(x) = λx and g i (x) = x + b i , where λ is a real-valued eigenvalue for A, and (β 1 , , β n ) is the associated eigenvector. It is known that any analytic action of Γ A on the circle is a ramified lift of one of the standard actions ρ λ . 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 Γ A on compact manifolds of higher dimension that are generated by C 1 diffeomorphisms close to the identity. We show that any action of Γ A 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 he r 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. Introduction 7 Chapter 2. Abelian-by-cyclic actions on S 1 15 2.1. Preliminaries 16 2.2. Global Invariant Sets 21 2.3. Local Analysis 28 2.4. The Liouville cocycle and BS(1,n) 38 2.5. Ramified Lifts of Affine Actions 41 2.6. Proof of Theorem 1.0.2 44 2.7. Unfaithful Actions 49 2.8. Actions Not Satisfying the Spectral Gap Condition 51 Chapter 3. Actions of Abelian-by-Cyclic Groups on Manifolds 53 3.1. Preliminaries 53 3.2. Invariant Sets for Actions of Γ A Close to the Identity 55 3.3. Rigidity for BS(1, n) close to id 57 3.4. Actions of Subgroups of Aff(R) on S 2 59 6 3.5. 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 R r (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 R r (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) = a, b|aba −1 = b n . The standard affine action of this group on the circle S 1  RP 1 given by ρ(a) : x → nx and ρ(b) : x → x + 1. This standard action can be used to generate many more actions by taking ramified lifts. An analytic map π :  M → M is said to be a ram ified covering map over the point p ∈ M, if the restriction of π to  M\{π −1 (p)} is a regular covering map onto M\{p}. A map  f :  M →  M is said to be a π-ramified lift of f : M → M if π ◦  f = f ◦ π for the ramified covering π. For any ramified covering map π : S 1 → S 1 there exists a π-ramified lift ρ n : BS(1, n) → Diff ω (S 1 ) of the standard representation, ρ n . 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 ρ n . 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, R r (Γ, M) carries a topology. For the group Γ, fix a generating set γ 1 , , γ k , and let d C r (ρ 1 , ρ 2 ) = sup γ 1 , ,γ k d C r (ρ 1 (γ i ), ρ 2 (γ i )). This topology is independent of the choice of generating set. Representations ρ, ρ  ∈ R r (Γ, M) are said to be conjugate in Diff r (M) if there exists h ∈ Diff r (M) such that h ◦ ρ(γ) = ρ  (γ) ◦ h for all γ ∈ Γ. A representation ρ 0 ∈ R r (Γ, M) is C (k,r) locally rigid 9 if there exists a C k -open neighborhood U of ρ 0 in R r (Γ, M) such that every ρ ∈ U is conjugate in Diff r (M) to ρ 0 . Let ρ 0 ∈ R ω (BS(1, n), S 1 ). [BW] show that there exists r ≥ 2 such that ρ 0 is C (1,r) locally rigid, but not C (1,r−1) locally rigid. The authors [BW] also give a classification of all analytic actions on S 1 of solvable groups. Theorem 1.0.1 (Burslem, Wilkinson). 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 Γ A = a, b 1 , b n |b i b j = b j b i , ab i a −1 =  j b a ij j . This group has a geometric interpretation as the fundamental group of the mapping cylinder for the map induced by A on T n . Note that in the case where A is the 1 × 1 matrix A = [n], we have that Γ A = 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 [...]... 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... 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... to actions of abelian-by-cyclic groups on higher dimensional manifolds In section 3.1 we summarize the local analysis of [B] concerning C 1 diffeomorphisms close to the identity Using these methods, we are able to prove Theorems 1.0.3 and 1.0.4, in sections 3.2 and 3.3, respectively We conclude this thesis with a catalog of examples of actions of solvable groups the sphere S 2 This is the content of. .. 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 It was later shown by Plante [P] that any action of a nilpotent Lie group on a surface with non-zero Euler characteristic has a global fixed point An example of Lima shows that it is not generally true that every action of a solvable Lie Group on... addresses actions of abelian-by-cyclic groups on the circle S 1 , and Chapter 3 investigates actions of abelianby-cyclic groups on higher dimensional manifolds In section 2.1 we discuss the primary background tools that will be needed in chapter 2 This includes Kopells’ Lemma and distortion estimates, and a specific application that is central to describing dynamical properties of actions of ΓA on S... 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 → λx and ρλ (bj ) : x → 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... D In particular, each component C of S c is component of Fix(gi ) for some i Each of these components is preserved by all gi , which proves (2) Proof of Property (3) Let C be a component of S c As above, we view C as a component of Fix(gi )c , for some i We may assume without loss of generality that gi is a contraction on C ∪ {p}, where p ∈ Fix(gi ) is an endpoint of C By Kopell’s lemma, any commuting... actions of ΓA on S 1 The proof of Theorem 1.0.2 requires two primary components The first is an analysis of compact globally invariant sets, and the second is a local characterization of faithful actions Section 2.2 contains the global analysis The main result of this section, Proposition 2.0.5, says that there is a set S that is a set of global fixed points for a finite index subgroup Γ < ΓA < Diff r (S 1... components are combined to complete the proof of Theorem 1.0.2 2.1 Preliminaries r We begin by introducing some essential background results We say that g ∈ Diff+ ([a, b)) is a contraction if g(a) = a and g(x) < x for all x ∈ (a, b) The following lemma of Kopell 17 about contractions is fundamental to the study of group actions in one dimension A nice description of this work is given in [N1] Lemma 2.1.1... process: consider the orbit of xi under gi+1 to find a point xi+1 that is fixed by gj for all j ≤ i + 1 It is clear that this process terminates We now consider the set of all common fixed points S = i Fix(gi ) This set S is the object that is characterized by Proposition 2.0.5 The analysis of this set will give a description of global invariant sets for actions of ΓA on S 1 Proof of Proposition 2.0.5 It . 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. Ramified Lifts of Affine Actions 41 2.6. Proof of Theorem 1.0.2 44 2.7. Unfaithful Actions 49 2.8. Actions Not Satisfying the Spectral Gap Condition 51 Chapter 3. Actions of Abelian-by-Cyclic Groups on. 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