Polyhedral CAT(0) metrics on locally finite complexes

TL;DR

This paper establishes that locally finite $CAT(0)$ complexes with convex vertex stars have arborescent structures, including cube and simplicial complexes, and characterizes when triangulated manifolds admit such metrics.

Contribution

It proves that $CAT(0)$ complexes with convex vertex stars are arborescent and characterizes $CAT(0)$ triangulated manifolds with polyhedral metrics.

Findings

Locally finite $CAT(0)$ complexes with convex vertex stars are arborescent.

$CAT(0)$ cube and equilateral simplicial complexes are arborescent.

Triangulated manifolds admit $CAT(0)$ polyhedral metrics iff they have arborescent triangulations.

Abstract

We prove the arborescence of any locally finite complex that is $CAT(0)$ with a polyhedral metric for which all vertex stars are convex. In particular locally finite $CAT(0)$ cube complexes or equilateral simplicial complexes are arborescent. Moreover, a triangulated manifold admits a $CAT(0)$ polyhedral metric if and only if it admits arborescent triangulations. We prove eventually that every locally finite complex which is $CAT(0)$ with a polyhedral metric has a barycentric subdivision which is arborescent.