Coarse $\mathcal{Z}$-Boundaries for Groups

TL;DR

This paper extends the concept of $ ext{Z}$-boundaries to a coarse setting, proving that key properties are preserved and establishing invariance under quasi-isometries, with new streamlined definitions and equivariant versions.

Contribution

It introduces the notion of coarse $ ext{Z}$-boundaries and model $ ext{Z}$-geometries, generalizing existing theory and proving invariance properties.

Findings

Coarse $ ext{Z}$-boundaries are quasi-isometry invariants.

Properties of $ ext{Z}$-boundaries extend to the coarse setting.

Streamlined definitions and equivariant versions are developed.

Abstract

We generalize Bestvina's notion of a $\mathcal{Z}$-boundary for a group to that of a "coarse $\mathcal{Z}$-boundary." We show that established theorems about $\mathcal{Z}$-boundaries carry over nicely to the more general theory, and that some wished-for properties of $\mathcal{Z}$-boundaries become theorems when applied to coarse $\mathcal{Z}$-boundaries. Most notably, the property of admitting a coarse $\mathcal{Z}$-boundary is a pure quasi-isometry invariant. In the process, we streamline both new and existing definitions by introducing the notion of a "model $\mathcal{Z}$-geometry." In accordance with the existing theory, we also develop an equivariant version of the above -- that of a "coarse $E\mathcal{Z}$-boundary."