Automorphisms of non-spherical buildings have unbounded displacement

TL;DR

The paper proves that nontrivial automorphisms of thick, purely infinite buildings cause unbounded displacement of chambers, leading to a trivial center for automorphism groups with bounded quotients.

Contribution

It establishes a new unbounded displacement property for automorphisms of non-spherical buildings and derives a group-theoretic consequence regarding the triviality of the center.

Findings

Nontrivial automorphisms have unbounded displacement.

Groups with bounded quotient have trivial center.

Results apply to thick buildings of purely infinite type.

Abstract

If f is a nontrivial automorphism of a thick building Delta of purely infinite type, we prove that there is no bound on the distance that f moves a chamber. This has the following group-theoretic consequence: If G is a group of automorphisms of Delta with bounded quotient, then the center of G is trivial.