Asymptotically rigid mapping class groups and Thompson's groups

TL;DR

This paper explores Thompson's groups as mapping class groups of infinite surfaces, introduces braided Thompson groups as extensions with braid groups, and discusses applications to quantization of Teichmüller spaces.

Contribution

It presents a novel perspective on Thompson's groups via infinite-type surface mapping class groups and introduces braided Thompson groups with applications to Teichmüller space quantization.

Findings

Introduction of braided Thompson groups as extensions of Thompson's groups

Connection between these groups and infinite surface mapping class groups

Applications to the quantization of Teichmüller spaces

Abstract

We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson's groups by infinite (spherical) braid groups. We will outline the main features of these groups and some applications to the quantization of Teichm\"uller spaces. The chapter provides an introduction to the subject with an emphasis on some of the authors results.