Flexibility of group actions on the circle

TL;DR

This paper develops a framework to produce uncountably many exotic, non-semi-conjugate actions of certain groups on the circle, revealing new flexibility phenomena and structural properties of these actions.

Contribution

It introduces a general framework for constructing uncountably many non-semi-conjugate circle actions for limit groups and Fuchsian groups, expanding understanding of group actions on the circle.

Findings

Uncountably many semi-conjugacy classes of faithful actions with disjoint rotation spectra.

Existence of actions not semi-conjugate to those factoring through finite-dimensional Lie groups.

General combination theorems for subgroups of PSL(2,R) and representations into topological groups.

Abstract

In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if $L$ is a nontrivial limit group then the nonlinear representation variety $\mathrm{Hom}(L,\mathrm{Homeo}_+(S^1))$ contains uncountably many semi-conjugacy classes of faithful actions on $S^1$ with pairwise disjoint rotation spectra (except for $0$) such that each representation lifts to $\mathbb{R}$. For the case of most Fuchsian groups $L$, we prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non-elementary free or surface group admits an action on $S^1$ that is never semi-conjugate to any action that factors through a finite--dimensional connected Lie subgroup in $\mathrm{Homeo}_+(S^1)$. It is exhibited that the mapping class groups of bounded surfaces have non-semi-conjugate faithful actions on $S^1$. In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of $\mathrm{PSL}_2(\mathbb{R})$ which apply to most Fuchsian groups and to all limit groups. We also show a Topological Baumslag Lemma, and general combination theorems for representations into Baire topological groups. The abundance of $\mathbb{Z}$--valued subadditive defect--one quasimorphisms on these groups would follow as a corollary. We also give a mostly self-contained reconciliation of the various notions of semi-conjugacy in the extant literature by showing that they are all equivalent.