Exotic mapping class group actions on the circle

TL;DR

This paper demonstrates that the standard faithful action of the mapping class group on the circle is not rigid, as finite index subgroups admit infinitely many non-conjugate faithful circle actions, answering a question by Farb.

Contribution

It shows the existence of infinitely many non-conjugate faithful circle actions for finite index subgroups of the mapping class group, revealing non-rigidity of the standard representation.

Findings

Existence of infinitely many non-conjugate faithful representations

Non-rigidity of the standard circle action of the mapping class group

Answers a question posed by Farb

Abstract

It has been known since the time of Nielsen that the mapping class group $\text{Mod}_{g,1}$ of a surface of genus $g$ and one puncture acts faithfully by homeomorphisms on the circle. In this note, we show that this standard representation of the mapping class group is not rigid, precisely, if $G<\text{Mod}_{g,1}$ is a finite index subgroup then there exist infinitely many non--conjugate faithful representations $G\to \text{Homeo}^+(S^1)$. We thus answer a question of B. Farb.