English

Exotic mapping class group actions on the circle

Geometric Topology 2016-10-18 v3 Group Theory

Abstract

It has been known since the time of Nielsen that the mapping class group Modg,1\text{Mod}_{g,1} of a surface of genus gg 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<Modg,1G<\text{Mod}_{g,1} is a finite index subgroup then there exist infinitely many non--conjugate faithful representations GHomeo+(S1)G\to \text{Homeo}^+(S^1). We thus answer a question of B. Farb.

Keywords

Cite

@article{arxiv.1603.02098,
  title  = {Exotic mapping class group actions on the circle},
  author = {Sang-hyun Kim and Thomas Koberda},
  journal= {arXiv preprint arXiv:1603.02098},
  year   = {2016}
}

Comments

This article has been subsumed by the preprint arXiv:1610.04098. This latter preprint also partially corrects a gap which appeared in this article

R2 v1 2026-06-22T13:05:19.851Z