Convex geodesic bicombings and hyperbolicity. (English) Zbl 1343.53036
Given a geodesic metric space \(X\), a geodesic bicombing maps each pair of points of \(X\) to a geodesic connecting them. The paper under review discusses and compares three different convexity notions of such geodesic bicombings – namely: convex and consistent, convex, conical. Busemann spaces, hence \(\mathrm{CAT}(0)\) spaces, always admit a geodesic bicombing which is convex and consistent. Also, for uniquely geodesic metric spaces, the three notions are equivalent. The authors prove that in metric spaces of finite combinatorial dimension in the sense of A. W. M. Dress [Adv. Math. 53, 321–402 (1984; Zbl 0562.54041)] convex geodesic bicombings are always unique and consistent (if they exist). This can be applied to the injective hull of a finitely generated Gromov hyperbolic group (endowed with the word metric) to construct a proper finite-dimensional metric space with a unique convex and consistent geodesic bicombing on which the group acts properly and cocompactly by isometries.
Reviewer: Jörg Lehnert (Leipzig)
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 20F65 | Geometric group theory |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
Citations:
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