A link condition for simplicial complexes, and CUB spaces

TL;DR

This paper introduces CUB spaces, a class of metric spaces with unique convex geodesic bicombings, generalizing nonpositive curvature notions, and provides a link condition to identify when simplicial complexes are CUB spaces.

Contribution

It establishes a link condition that characterizes when simplicial complexes with polyhedral metrics are CUB spaces, extending Gromov's link condition.

Findings

The link condition applies to Euclidean buildings and Artin complexes.

CUB spaces include CAT(0) and Busemann-convex spaces.

Several examples like Garside groups and curve complexes satisfy the link condition.

Abstract

We motivate the study of metric spaces with a unique convex geodesic bicombing, which we call CUB spaces. These encompass many classical notions of nonpositive curvature, such as CAT(0) spaces and Busemann-convex spaces. Groups having a geometric action on a CUB space enjoy numerous properties. We want to know when a simplicial complex, endowed with a natural polyhedral metric, is CUB. We establish a link condition, stating essentially that the complex is locally a lattice. This generalizes Gromov's link condition for cube complexes, for the $\ell^\infty$ metric. The link condition applies to numerous examples, including Euclidean buildings, simplices of groups, Artin complexes of Euclidean Artin groups, (weak) Garside groups, some arcs and curve complexes, and minimal spanning surfaces of knots.