Groups acting on spaces of non-positive curvature

TL;DR

This survey explores groups acting on non-positively curved metric spaces, discussing key examples, their rigidity, and the relationship between algebraic, analytic, and geometric properties, along with open conjectures.

Contribution

It provides a comprehensive overview of the main examples, properties, and conjectures related to groups acting on non-positive curvature spaces.

Findings

Identification of key examples of non-positively curved spaces

Connections between group properties and geometric rigidity

Presentation of open conjectures in the field

Abstract

In this survey article, we present some panorama of groups acting on metric spaces of non-positive curvature. We introduce the main examples and their rigidity properties , we show the links between algebraic or analytic properties of the group and geometric properties of the space. Finally, we conclude with a few conjectures in the subject.