Dimension rigidity of lattices in semisimple Lie groups

TL;DR

This paper proves that for lattices in isometry groups of certain symmetric spaces, their virtual cohomological dimension matches their proper geometric dimension, clarifying their algebraic and geometric properties.

Contribution

It establishes the equality of virtual cohomological and geometric dimensions for lattices in non-compact symmetric spaces, a significant step in understanding their structure.

Findings

Virtual cohomological dimension equals geometric dimension for these lattices.

Results apply to lattices in symmetric spaces without Euclidean factors.

Provides new insights into the rigidity and structure of such lattices.

Abstract

We prove that if $\Gamma$ is a lattice in the group of isometries of a symmetric space of non-compact type without euclidean factors, then the virtual cohomological dimension of $\Gamma$ equals its proper geometric dimension.