# Hyperbolic isometries and boundaries of systolic complexes

## TL;DR

This paper investigates how hyperbolic isometries act on the boundaries of systolic complexes, revealing fixed points, conditions for trivial action, and introducing a generalized displacement set to analyze these actions.

## Contribution

It introduces the concept of a $K$-displacement set for hyperbolic isometries and characterizes their boundary actions, also proving systolic complexes are almost extendable under group actions.

## Key findings

- Hyperbolic isometries have canonical fixed points on the boundary.
- An isometry acts trivially on the boundary iff it is virtually central.
- Systolic complexes with group actions are almost extendable.

## Abstract

Given a group $G$ acting geometrically on a systolic complex $X$ and a hyperbolic isometry $h \in G$, we study the associated action of $h$ on the systolic boundary $\partial X$. We show that $h$ has a canonical pair of fixed points on the boundary and that it acts trivially on the boundary if and only if it is virtually central. The key tool that we use to study the action of $h$ on $\partial X$ is the notion of a $K$-displacement set of $h$, which generalises the classical minimal displacement set of $h$. We also prove that systolic complexes equipped with a geometric action of a group are almost extendable.