Contracting Boundaries of CAT(0) Spaces

TL;DR

The paper introduces the contracting boundary for CAT(0) spaces, a new quasi-isometry invariant based on hyperbolic-like rays, helping distinguish quasi-isometry classes of certain groups.

Contribution

It defines the contracting boundary, proves its invariance under quasi-isometries, and applies it to differentiate classes of right-angled Coxeter groups.

Findings

Contracting boundary is a quasi-isometry invariant.

All five hyperbolic-like properties are equivalent.

Used to distinguish quasi-isometry classes of specific groups.

Abstract

As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like properties. We prove that these properties are all equivalent and that the contracting boundary is a quasi-isometry invariant. We use this invariant to distinguish the quasi-isometry classes of certain right-angled Coxeter groups.