Thompson's groups and Teichmüller modular groups of generalized Cantor sets

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 discuss Thompson's groups $F,T$ and $V$ in Teichmüller modular groups of Teichmüller spaces of generalized Cantor sets with certain conditions by using a quasiconformal geometric method. Moreover, we show that groups corresponding to $F$ and $T$ act properly discontinuously on such Teichmüller spaces. We also show the same results for the asymptotic Teichmüller space. E. de Faria, F. P. Gardiner and W. J. Harvey show similar results on Thompson's group $F$ and asymptotic Teichmüller spaces. So, our results are improvement of them. We also discuss the actions of Thompson's groups on other spaces which we call \lq\lq pre-Tiechmüller spaces" of generalized Cantor sets. Especially, we show that the action of a group which is isomorphic to Thompson's group $V$ on the pre-Teichüller space is not properly discontinuous while groups isomorphic to $F$ and $T$ act properly discontinuously. Finally, we construct infinitely many distinct Teichmüller spaces of generalized Cantor sets where our results can apply.