Structure and regularity of group actions on one-manifolds

TL;DR

This paper explores how the algebraic properties of finitely generated groups influence their actions on one-dimensional manifolds, providing a comprehensive classification of regularity and obstructions to smoothness.

Contribution

It offers a uniform construction of finitely generated groups with prescribed regularity on intervals and circles, and develops a theory of dynamical obstructions to smoothness.

Findings

Classification of right-angled Artin groups by finite critical regularity

Computation of virtual critical regularity for most mapping class groups

Development of a theory of dynamical obstructions to smoothness

Abstract

In this monograph, we give an account of the relationship between the algebraic structure of finitely generated and countable groups and the regularity with which they act on manifolds. We concentrate on the case of one--dimensional manifolds, culminating with a uniform construction of finitely generated groups acting with prescribed regularity on the compact interval and on the circle. We develop the theory of dynamical obstructions to smoothness, beginning with classical results of Denjoy, to more recent results of Kopell, and to modern results such as the $abt$--Lemma. We give a classification of the right-angled Artin groups that have finite critical regularity and discuss their exact critical regularities in many cases, and we compute the virtual critical regularity of most mapping class groups of orientable surfaces.