$\ell^p$ metrics on cell complexes
Thomas Haettel, Nima Hoda, Harry Petyt
TL;DR
We study cell complexes endowed with piecewise $\ell^p$ metrics as a flexible interpolation between $\ell^1$ (median/injective) and $\ell^2$ (CAT(0)) geometries. A local-to-global gluing criterion is developed to deduce $Busemann$-convexity and unique geodesicity, and these ideas yield strong bolicity under mild hypotheses. In CAT(0) cube complexes, we show $Busemann$-convexity for all $p$, with uniform convexity and uniform smoothness for $p\ge 2$, and establish a robust local description of geodesics via a zero-tension and no-shortcut framework, plus an explicit distance formula. We further connect these geometric properties to group actions, proving centraliser splitting and deriving corollaries for mapping class groups, demonstrating the utility of $\ell^p$ metrics in geometric group theory. Collectively, the work provides a versatile toolkit for analyzing nonpositive curvature and geodesic structure across a spectrum of $\ell^p$ geometries 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 nonpositive 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.