Hyperbolic groups with $k$-geodetic Cayley graphs

Abstract: A locally-finite simple connected graph is said to be $k$-geodetic for some $k\geq1$, if there is 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 generalise 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.

group theory

Audience: researchers in the discipline

Symmetry in Newcastle