2022/11/24 by Elder, Murray, Piggott, Adam, Townsend, Kane · 1 citation
#20F65 (Primary) 5C12 #20F67 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2211.13397
We call a graph k-geodetic, for some k≥ 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.