  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.
