# Geodesic rays, the "Lion-Man" game, and the fixed point property

## TL;DR

This paper explores the connections between geodesic rays, fixed point properties, and a lion-and-man game within Gromov hyperbolic CAT(0) spaces, revealing new insights into geometric and game-theoretic properties.

## Contribution

It establishes a novel relationship among geodesic rays, fixed point properties, and pursuit-evasion games specifically in Gromov hyperbolic CAT(0) spaces.

## Key findings

- Existence of geodesic rays relates to fixed point properties.
- Characterization of the lion-man game in hyperbolic spaces.
- New links between geometric curves and pursuit-evasion dynamics.

## Abstract

This paper focuses on the relation among the existence of different types of curves (such as directional ones, quasi-geodesic or geodesic rays), the (approximate) fixed point property for nonexpansive mappings, and a discrete lion and man game. Our main result holds in the setting of CAT(0) spaces that are additionally Gromov hyperbolic.