Twisted conjugacy and quasi-isometric rigidity of irreducible lattices in semisimple Lie groups

TL;DR

This paper proves that groups quasi-isometric to irreducible lattices in semisimple Lie groups, as well as the lattices themselves, have infinitely many twisted conjugacy classes for all automorphisms, revealing a strong rigidity property.

Contribution

It establishes the $R_ fty$-property for groups quasi-isometric to and including lattices in semisimple Lie groups, extending previous results and demonstrating a form of quasi-isometric rigidity.

Findings

Groups quasi-isometric to irreducible lattices have the $R_ Infty$-property.

Lattices in semisimple Lie groups also have the $R_ Infty$-property.

Extension of earlier results to broader classes of lattices.

Abstract

Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $\Gamma$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely, that there are infinitely $\phi$-twisted conjugacy classes for every automorphism $\phi$ of $\Gamma$. Also, we show that any lattice in $G$ has the $R_\infty$-property, extending our earlier result for irreducible lattices.