Weak Tits alternative for groups acting geometrically on buildings

TL;DR

This paper proves that groups acting geometrically on buildings are either virtually abelian or contain a free subgroup, extending the understanding of group actions in geometric group theory.

Contribution

It establishes a weak Tits alternative for groups acting on buildings, a significant generalization in geometric group theory.

Findings

Groups acting on buildings are either virtually abelian or contain free subgroups.

The result applies to type-preserving automorphisms of buildings.

Provides new insights into the structure of groups acting on geometric spaces.

Abstract

We show that if a group $G$ acts geometrically by type-preserving automorphisms on a building, then $G$ satisfies the weak Tits alternative, namely, that $G$ is either virtually abelian or contains a non-abelian free group.