Injective Metrics on Cube Complexes

Benjamin Miesch

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arXiv · 1411.7234

Injective Metrics on Cube Complexes

Benjamin Miesch

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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.

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Benjamin Miesch. 2014-11-26. Injective Metrics on Cube Complexes. https://arxiv.org/abs/1411.7234

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