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.

Sectional Curvature-Type Conditions on Metric Spaces

“…This property resembles Kleiner's notion of often convex spaces [KL97], resp. the notion of convex bicombings [DL14].…”

On the geometry and fixed point free mappings in hyperbolic spaces

“…In [4,Lemma 5.3], it was proved that this definition is independent of the choice of a base point, and so is well defined.…”

Majorization by hemispheres and quadratic isoperimetric constants

“…If X is a Banach space one may define a contracting barycenter map simply via b(µ) ∶= ∫ X x dµ(x) and if X is a Hadamard space by minimizing the functional q ↦ ∫ X d 2 (p, q)dµ(p). It turns out that contracting barycenter maps have a more geometric equivalent which are conical bicombings introduced by Dominic Descombes and Urs Lang in [DL15]. A conical (geodesic) bicombing σ on X is a map assigning to every tupel of points (x, y) in X a geodesic σ x,y such that for any pair of tupels (x, y), (x ′ , y ′ ) the distance function between σ x,y and σ x ′ ,y ′ satisfies a weak convexity condition, see section 2.2.…”