WoG 2022 Talk 1.1: Kane Townsend - Hyperbolic groups with k-geodetic Cayley graph

Speaker: Kane Townsend (University of Technology Sydney)

Title: Hyperbolic groups with $k$-geodetic Cayley graph
Abstract: A locally-finite simple connected graph is said to be \emph{$k$-geodetic} for some $k\geq 1$, if there is at most $k$ geodesic paths between any two vertices. We investigate the properties of hyperbolic groups that have $k$-geodetic Cayley graphs. We start by showing that a $k$-geodetic graph cannot have a ``ladder-like structure" with unbounded length. Adding the assumption that the $k$-geodetic graph is the Cayley graph of a hyperbolic group we generalise a well-known result, proven by Papasoglu, from $k=1$ to all $k\geq 1$. We conjecture that groups with $k$-geodetic Cayley graph are exactly the same class of groups as those with $1$-geodetic Cayley graph.