Thompson's groups and Teichm\"uller modular groups of generalized Cantor sets

TL;DR

This paper explores the relationship between Thompson's groups and Teichmüller modular groups of generalized Cantor sets, revealing their subgroup structures and action properties on Teichmüller spaces.

Contribution

It establishes Thompson's groups as subgroups of Teichmüller modular groups and analyzes their actions on Teichmüller spaces of generalized Cantor sets.

Findings

Thompson's groups are subgroups of Teichmüller modular groups.

F and T act properly discontinuously on these spaces.

V does not act properly discontinuously.

Abstract

Thompson's groups, which are denoted by $F, T$ and $V$, were introduced by R. Thompson. It is known that they are related to various fields in mathematics. In this paper, we establish that Thompson's groups are regarded as subgroups of Teichm\"uller modular groups of Teichm\"uller spaces of generalized Cantor sets. Moreover, Thompson's groups $F$ and $T$ act properly discontinuously on such Teichm\"uller spaces but Thompson's group $V$ does not. In some sense, those results are improvements of the results by E. de Faria, F. P. Gardiner and W. J. Harvey on Thomnpson's group $F$ and asymptotic Teichm\"uller spaces. We also show that Thompson's groups act infinitely many Teichm\"uller spaces of generalized Cantor sets.