Some properties of non-positively curved lattices

TL;DR

This paper investigates the structure and properties of non-positively curved lattices, focusing on CAT(0) groups and spaces, extending classical results to singular spaces and establishing new rigidity and arithmeticity theorems.

Contribution

It provides new insights into the structure of CAT(0) groups and spaces, including generalizations of classical theorems to singular spaces and novel rigidity results.

Findings

General study of isometry groups of proper CAT(0) spaces

Extension of classical Hadamard manifold results to singular spaces

New arithmeticity and rigidity theorems for CAT(0) lattices

Abstract

We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity statements are obtained.