# On geodesic ray bundles in hyperbolic groups

## TL;DR

This paper constructs a specific Cayley graph of a hyperbolic group to demonstrate that certain geodesic ray bundles can differ infinitely, answering a question about their structure at infinity.

## Contribution

It provides a counterexample showing that geodesic ray bundles in hyperbolic groups can have infinite symmetric difference, addressing an open question.

## Key findings

- Existence of elements with infinitely different geodesic ray bundles
- Counterexample to previous assumptions about geodesic rays
- Clarification of boundary behavior in hyperbolic groups

## Abstract

We construct a Cayley graph $\mathbf{Cay}_S(\Gamma)$ of a hyperbolic group $\Gamma$ such that there are elements $g,h\in\Gamma$ and a point $\gamma \in \partial_\infty\Gamma = \partial_\infty\mathbf{Cay}_S(\Gamma)$ such that the sets $\mathcal{RB}(g,\gamma)$ and $\mathcal{RB}(h,\gamma)$ in $\mathbf{Cay}_S(\Gamma)$ of vertices along geodesic rays from $g,h$ to $\gamma$ have infinite symmetric difference; thus answering a question of Huang, Sabok and Shinko.