Global Rigidity of Some Abelian-by-Cyclic group actions on $\T^2$

TL;DR

This paper classifies certain group actions on the 2-torus involving Abelian-by-Cyclic groups, establishing conditions for topological and smooth conjugacy, and proves a Tits alternative type theorem for these groups.

Contribution

It provides a complete classification of Abelian-by-Cyclic group actions on $ ^2$ up to conjugacy and introduces a Tits alternative type theorem for these groups.

Findings

Complete classification of group actions on $ ^2$

Conditions for topological and smooth conjugacy

A Tits alternative type theorem for diffeomorphism groups

Abstract

For groups of diffeomorphisms of $\T^2$ containing an Anosov diffeomorphism, we give a complete classification for polycyclic Abelian-by-Cyclic group actions on $\T^2$ up to both topological conjugacy and smooth conjugacy under mild assumptions. Along the way, we also prove a Tits alternative type theorem for some groups of diffeomorphisms of $\T^2$.