ℓ p $\ell ^p$ metrics on cell complexes

Abstract

Motivated by the observation that groups can be effectively studied using metric spaces modelled on ${\ell }^1$, ${\ell }^2$ and ${\ell }^{\infty }$ geometry, we consider cell complexes equipped with an ${\ell }^p$ metric for arbitrary $p$. Under weak conditions that can be checked locally, we establish non-positive curvature properties of these complexes, such as Busemann-convexity and strong bolicity. We also provide detailed information on the geodesics of these metrics in the special case of CAT(0) cube complexes.