Injective Metrics on Cube Complexes

TL;DR

This paper characterizes when finite-dimensional CAT(0) cube complexes admit injective metrics, showing they are always CAT(0) if injective, and provides criteria for such metrics to exist, along with related geodesic properties.

Contribution

It establishes that injective metrics on cube complexes imply CAT(0) geometry and offers a criterion for finite-dimensional complexes with finite width to have injective metrics.

Findings

Injective metrics on cube complexes imply they are CAT(0).

Finite-dimensional CAT(0) cube complexes with finite width can admit injective metrics.

Cube complexes with $l_p$-norms are geodesic.

Abstract

For locally finite CAT(0) cube complexes it is known that they are injectively metrizable choosing the $l_\infty$-norm on each cube. In this paper we show that cube complexes which are injective with respect to this metric are always CAT(0). Moreover we give a criterion for finite dimensional CAT(0) cube complexes with finite width to posses an injective metric. As a side result we prove a modification of Bridson's Theorem for cube complexes saying that finite dimensional cube complexes with $l_p$-norms on the cubes are geodesic.