Isometry groups of non-positively curved spaces: discrete subgroups

TL;DR

This paper investigates the properties of lattices in non-positively curved spaces, establishing key results like Borel density, Mostow rigidity, and characterizations of symmetric spaces, advancing understanding of geometric group theory.

Contribution

It provides new results on Borel density, Mostow rigidity, and characterizations of symmetric spaces within non-positively curved metric spaces, including residual finiteness and arithmeticity of lattices.

Findings

Borel density established for lattices in non-positively curved spaces

A form of Mostow rigidity proved in this setting

Characterization of symmetric spaces among CAT(0) spaces

Abstract

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.