The Structure of Homeomorphism and Diffeomorphism Groups

Mann, Kathryn

doi:10.1090/noti2252

Cited by 1 publication

(1 citation statement)

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“…Why, after all, should there be no way to find a simultaneous smoothing of all C k diffeomorphisms of M ? The answer is that one cannot, and we will return to this question in Section 3 below; the reader may also consult [109,148,156,157,158,201,208] for some background results in this direction.…”

Section: Groups Of Manifold Diffeomorphismsmentioning

confidence: 99%

Structure and regularity of group actions on one-manifolds

Kim¹,

Koberda²

2021

Preprint

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This book represents an account and contextualization of a research program that was executed by the authors and their collaborators over the period of several years. Both authors began their careers in classical geometric group theory and began collaborating on projects about right-angled Artin groups and their relationship with mapping class groups of surfaces, at a time when the first author was a visiting assistant professor at Tufts University and while the second author was a graduate student at Harvard University.Around 2014, the authors became interested in a question attributed by M. Kapovich to V. Kharlamov, one that asks which right-angled Artin groups can act faithfully by smooth diffeomorphisms on the circle. Because of the profusion of right-angled Artin subgroups of mapping class groups and because of the robust persistence of right-angled Artin groups under passing to finite index subgroups, an answer to Kharlamov's question had the potential to shed light on a question of F. Labourie that he posed in his 1998 ICM talk and that was reiterated by B. Farb and J. Franks in several places: are there finite index subgroups of (sufficiently complicated) mapping class groups acting smoothly on the circle?At first, it seemed to the authors that an easy solution to Kharlamov's question was available, and that all right-angled Artin groups admitted such faithful actions. It was soon pointed out by J. Bowden that the actions constructed by the authors and H. Baik in fact failed to be differentiable at certain accumulations of fixed points, and so only furnished faithful smooth actions on the real line. After approximately two years of work, inspired by Bowden's observation, Baik and the authors managed to prove that that most right-angled Artin groups do not admit faithful C 2 actions on the compact interval nor on the circle. As a consequence, Labourie's question could be completely answered in the case of C 2 actions of finite index subgroups of mapping class groups.About a year and a half later, the authors were able to develop a tool that they dubbed the abt-Lemma, which is a certain combinatorial-algebraic obstruction for smoothness of a group action on a compact one-manifold. As a result, they were able to answer Kharlamov's question by giving a concise characterization of rightangled Artin groups acting faithfully by smooth diffeomorphisms on the circle.Because of the technical details involved in the analysis of smooth actions of right-angled Artin groups on one-manifolds, the authors became interested in Thompson's group F , which occurs naturally in these contexts. Inspired by the dynamical theory of Thompson's group, the authors investigated the class of chain groups in subsequent work with Y. Lodha. These objects form a highly diverse class of groups of homeomorphisms of the interval, while nevertheless exhibiting remarkable uniformity properties. i v Contents Preface i Acknowledgements v Chapter 1. Introduction Appendix B. Orderability and Hölder's Theorem 1. Orderability of groups and homeomorp...

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