On uniform lattices in real semisimple groups

TL;DR

This paper proves that co-compactness of arithmetic lattices in connected semisimple real Lie groups is preserved under representation equivalence, extending understanding of lattice properties in Lie groups.

Contribution

It establishes that co-compactness is invariant under representation equivalence for arithmetic lattices in semisimple real Lie groups.

Findings

Co-compactness is preserved under representation equivalence.

Extends results related to weakly commensurable subgroups.

Provides insights into the structure of arithmetic lattices.

Abstract

In this article we prove that the co-compactness of the arithmetic lattices in a connected semisimple real Lie group is preserved if the lattices under consideration are representation equivalent. This is in the spirit of the question posed by Gopal Prasad and A. S. Rapinchuk where instead of representation equivalence, the lattices under consideration are weakly commensurable Zariski dense subgroups.