On $k$-geodetic graphs and groups

TL;DR

This paper proves that hyperbolic groups with $k$-geodetic Cayley graphs are virtually-free and have cyclic centralisers for infinite order elements, extending known results from the case $k=1$ using new graph theoretic methods.

Contribution

It generalizes previous results on hyperbolic groups with 1-geodetic graphs to arbitrary $k$, introducing new ladder-like structure analysis in $k$-geodetic graphs.

Findings

Hyperbolic groups with $k$-geodetic Cayley graphs are virtually-free.

Centralisers of infinite order elements are infinite cyclic.

New graph theoretic results on ladder-like structures in $k$-geodetic graphs.

Abstract

We call a graph $k$-geodetic, for some $k\geq 1$, if it is connected and between any two vertices there are at most $k$ geodesics. It is shown that any hyperbolic group with a $k$-geodetic Cayley graph is virtually-free. Furthermore, in such a group the centraliser of any infinite order element is an infinite cyclic group. These results were known previously only in the case that $k=1$. A key tool used to develop the theorem is a new graph theoretic result concerning ``ladder-like structures'' in a $k$-geodetic graph.