Twisted Conjugacy Classes in Lattices in Semisimple Lie Groups

TL;DR

This paper proves that irreducible lattices in higher-rank semisimple Lie groups have infinitely many twisted conjugacy classes for any automorphism, demonstrating a strong rigidity property.

Contribution

It establishes the $R_ fty$-property for irreducible lattices in semisimple Lie groups of rank at least 2, extending understanding of conjugacy class behavior.

Findings

Irreducible lattices in higher-rank semisimple Lie groups have the $R_ extinfty$-property.

The $R_ extinfty$-property holds for all automorphisms of these lattices.

This property indicates a form of algebraic rigidity in such groups.

Abstract

Given a group automorphism $\phi:\Gamma\to \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-conjugacy classes. One says that $\Gamma$ has the $R_\infty$-property if there are infinitely many $\phi$-conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this paper we show that any irreducible lattice in a connected semi simple Lie group having finite centre and rank at least 2 has the $R_\infty$-property.