A locally-finite simple connected graph is said to be k-geodetic for some k ≥ 1, if there are at most k distinct geodesics between any two vertices of the graph. We investigate the properties of hyperbolic groups with k-geodetic Cayley graphs. To begin, we show that k-geodetic graphs cannot have a “ladder-like” geodesic structure with unbounded length. Using this bound, we generalize a well-known result of Papasoglu that states hyperbolic groups with 1-geodetic Cayley graphs are virtually free. We then investigate which elements of the hyperbolic group with k-geodetic Cayley graph commute with a given infinite order element.
This video was produced by Newcastle University, Australia, as part of the Symmetries in Newcastle seminar series.
