Coarse median algebras: The intrinsic geometry of coarse median spaces and their intervals

TL;DR

This paper introduces a new algebraic framework for coarse median spaces, connecting asymptotic geometry, algebra, and combinatorics, and explores their properties through intervals and rank concepts.

Contribution

It develops a coarse median algebra concept that generalizes existing structures and introduces a notion of rank, linking to hyperbolicity and geometric properties.

Findings

Metric can be reconstructed from the coarse median operator up to quasi-isometry.

Finite rank coarse median algebras serve as higher-dimensional analogues of hyperbolic spaces.

Intervals' cardinality effectively proxies for distance in coarse median spaces.

Abstract

This paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov's concept of $\delta$-hyperbolicity.