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

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