Coarse Z-Boundaries for Groups

Abstract

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