Convex geodesic bicombings and hyperbolicity

Abstract

A geodesic bicombing on a metric space selects for every pair of points a geodesic connecting them. We prove existence and uniqueness results for geodesic bicombings satisfying different convexity conditions. In combination with recent work by the second author on injective hulls, this shows that every word hyperbolic group acts geometrically on a proper, finite dimensional space 𝑋 with a unique (hence equivariant) convex geodesic bicombing of the strongest type. Furthermore, the Gromov boundary of 𝑋 is a 𝑍-set in the closure of 𝑋, and the latter is a metrizable absolute retract, in analogy with the Bestvina–Mess theorem on the Rips complex.