Flexibility of Group Actions on the Circle

verfasst von: Assoc. Prof. Sang-hyun Kim, Assist. Prof. Thomas Koberda, Prof. Mahan Mj

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

In this partly expository work, a framework is developed for building exotic circle actions of certain classical groups.

The authors give general combination theorems for indiscrete isometry groups of hyperbolic space which apply to Fuchsian and limit groups. An abundance of integer-valued subadditive defect-one quasimorphisms on these groups follow as a corollary.

The main classes of groups considered are limit and Fuchsian groups. Limit groups are shown to admit large collections of faithful actions on the circle with disjoint rotation spectra. For Fuchsian groups, further flexibility results are proved and the existence of non-geometric actions of free and surface groups is established. An account is given of the extant notions of semi-conjugacy, showing they are equivalent.

This book is suitable for experts interested in flexibility of representations, and for non-experts wanting an introduction to group representations into circle homeomorphism groups.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In this monograph, we study finitely generated groups which are classically known to act faithfully on the circle. The purpose of this monograph is to give a systematic construction of uncountable families of actions of these groups which have “essentially different” dynamics. The tools described allow us to construct many exotic actions of classically studied groups, i.e. actions which are not semi-conjugate to the “usual” or “standard” actions of these groups. This monograph is partially expository and partially original. We develop theory as coherently as possible, with some methods that are well-known to experts, and others which to our knowledge are our own.

Sang-hyun Kim, Thomas Koberda, Mahan Mj

Chapter 2. Preliminaries

Abstract

In this chapter, we briefly review well-known facts on circle actions and on (discrete or indiscrete) subgroups of $ \operatorname {\mathrm {PSL}}_2(\mathbb {R})$.

Sang-hyun Kim, Thomas Koberda, Mahan Mj

Chapter 3. Topological Baumslag Lemmas

Abstract

This chapter deals with one of the principal technical tools of the monograph, namely an extension of Baumslag’s Lemma. The following are some of the main ingredients in this chapter:

A Topological Baumslag Lemma, which gives sufficient conditions to guarantee nontriviality of a word

$$\displaystyle w (t_1,\cdots , t_k) = g_1 \mu _1(t_1) \cdots g_k \mu _k (t_k) $$

for large values of the parameters t _i. Here the μ _j(t _j)s are one-parameter subgroups of a continuous (possibly analytic) group. The proof is quite general and reminiscent of Tits’ proof (Tits, J Algebra 20(2):250–270, 1972) of the Tits’ alternative for discrete linear groups in that the underlying idea consists of a ping-pong argument.

The one-parameter subgroups are often (generalizations of) parabolic and hyperbolic one-parameter subgroups of $ \operatorname {\mathrm {PSL}}_2(\mathbb {R})$. To handle elliptic subgroups we use a complexification and Zariski density trick by embedding $ \operatorname {\mathrm {PSL}}_2(\mathbb {R})$ in $ \operatorname {\mathrm {PSL}}_2(\mathbb {C})$ and reduce the elliptic case to the hyperbolic one (Lemma 3.4).

Sang-hyun Kim, Thomas Koberda, Mahan Mj

Chapter 4. Splittable Fuchsian Groups

Abstract

In this chapter, we study deformations of (possibly indiscrete) faithful representations of Fuchsian groups such that almost all points on the deformations are still faithful. Let L be a splittable Fuchsian group, which includes all Fuchsian groups with Euler characteristic at most − 1; see Definition 4.3. We will prove that an arbitrarily small deformation of a given representation can be chosen so that the new trace spectrum is almost disjoint from the original one (Theorem 4.1). Then we show X _proj(L) contains at least one indiscrete representation (Lemma 4.10). Moreover, if an open set U contains at least one indiscrete representation in X _proj(L), then U contains uncountably many pairwise inequivalent indiscrete representation in X _proj(L) (Theorem 4.2).

Sang-hyun Kim, Thomas Koberda, Mahan Mj

Chapter 5. Combination Theorem for Flexible Groups

Abstract

In this chapter, we establish a combination theorem (Theorem 5.1) for the class of flexible and liftable-flexible groups. This generalizes the arguments in Chap. 4 that most Fuchsian groups are flexible. Implications of flexibility and liftability for limit groups and quasimorphisms are discussed here.

Sang-hyun Kim, Thomas Koberda, Mahan Mj

Chapter 6. Axiomatics

Abstract

In this chapter, we develop a more axiomatic setup for the discussion in Chaps. 4 and 5, and establish indiscrete combination theorems again for all limit groups and in the setup where the target group is a general Baire topological group.

Sang-hyun Kim, Thomas Koberda, Mahan Mj

Chapter 7. Mapping Class Groups

Abstract

In this chapter, we shift the focus to exotic actions of mapping class groups on the circle, where here “exotic” means “not conjugate to Nielsen’s standard action” (see Handel and Thurston (Adv Math 56:173–191, 1985) and Casson and Bleiler (Automorphisms of Surfaces After Nielsen and Thurston, Cambridge University Press, Cambridge, 1988) for detailed discussions of Nielsen’s action). We first discuss actions of fibered, hyperbolic 3-manifold groups on S ¹, in relation to Nielsen’s action.

Sang-hyun Kim, Thomas Koberda, Mahan Mj

Chapter 8. Zero Rotation Spectrum and Teichmüller Theory

Abstract

In this chapter, we consider free group and surface group actions on the circle, and develop conditions under which the equivalence class of an action is determined by the rotation spectrum, and when the semi-conjugacy class of the action is determined by the marked rotation spectrum. In the case of indiscrete representations of groups into $ \operatorname {\mathrm {PSL}}_2(\mathbb {R})$, there is a lack of a geometric interpretation of such representations which is as well-developed as Teichmüller theory in the case of discrete representations. We will consider the degree to which marked rotation spectrum can supplant marked length spectrum as a (sometimes nearly complete) semi-conjugacy invariant.

Sang-hyun Kim, Thomas Koberda, Mahan Mj

Backmatter

Titel: Flexibility of Group Actions on the Circle
verfasst von: Assoc. Prof. Sang-hyun Kim
Assist. Prof. Thomas Koberda
Prof. Mahan Mj
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-02855-8
Print ISBN: 978-3-030-02854-1
DOI: https://doi.org/10.1007/978-3-030-02855-8