Injective Hulls of Infinite Totally Split-Decomposable Metric Spaces

We extend the theory of splits in finite metric spaces to infinite ones. Within this more general framework, we investigate the class of spaces having metrics that are integer-valued and totally split-decomposable, as well as the polyhedral complex structure of their injective hulls. For this class, we provide a characterization for the injective hull to be combinatorially equivalent to a CAT(0) cube complex. Intermediate results include the generalization of the decomposition theory introduced by Bandelt and Dress in 1992 as well as results on the tight span of totally split-decomposable metric spaces proved by Huber, Koolen, and Moulton in 2006. Next, using results of Lang from 2013, we obtain proper actions on CAT(0) cube complexes for finitely generated groups endowed with a totally split-decomposable word metric and for which the associated splits satisfy a simple combinatorial property. In the case of Gromov hyperbolic groups, the obtained action is both proper aand co-compact. Finally, we obtain as an application that injective hulls of odd cycles are cell complexes isomorphic to CAT(0) cube complexes.