Minimal displacement set for weakly systolic complexes

TL;DR

This paper studies the minimal displacement set in weakly systolic complexes, showing it is systolic and embeds isometrically, and classifies isometries into elliptic or hyperbolic types based on their action.

Contribution

It introduces the structure of the minimal displacement set in weakly systolic complexes and characterizes isometries, providing new insights into their geometric behavior.

Findings

Minimal displacement set is systolic.

Isometries either fix a barycentre or stabilize a thick geodesic.

Embedding of the minimal displacement set is isometric.

Abstract

We investigate the structure of the minimal displacement set in weakly systolic complexes. We show that such set is systolic and that it embeds isometrically into the complex. As corollaries, we prove that any isometry of a weakly systolic complex either fixes the barycentre of some simplex (elliptic case) or it stabilizes a thick geodesic (hyperbolic case).