Injective Hulls of Infinite Totally Split-Decomposable Metric Spaces

TL;DR

This paper characterizes when the injective hull of infinite totally split-decomposable metric spaces is a CAT(0) cube complex, extending decomposition theory and applying it to group actions on such complexes.

Contribution

It provides a new characterization linking injective hulls of certain metric spaces to CAT(0) cube complexes, extending existing decomposition theories.

Findings

Injective hulls can be polyhedral complexes.

Characterization of when these hulls are CAT(0) cube complexes.

Proper and cocompact group actions on these complexes.

Abstract

We consider the class of (possibly) infinite metric spaces with integer-valued totally split-decomposable metric and possessing an injective hull which has the structure of a polyhedral complex. For this class, we give a characterization for the injective hull to be combinatorially equivalent to a CAT(0) cube complex. In order to obtain these results, we extend 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. As an application, and using results of Lang of 2013, we obtain proper actions on CAT(0) cube complexes for finitely generated groups endowed with a totally split-decomposable word metric whose associated splits satisfy an easy combinatorial property. In the case of Gromov hyperbolic groups, the action is proper as well as cocompact.