Boundaries of Baumslag-Solitar Groups

TL;DR

This paper investigates the boundary structures of Baumslag-Solitar groups, demonstrating that all such groups admit EZ-structures and generalized ones admit Z-structures, extending the understanding of group boundaries.

Contribution

It proves that all Baumslag-Solitar groups admit EZ-structures and all generalized Baumslag-Solitar groups admit Z-structures, expanding the classes of groups known to have these boundary structures.

Findings

All Baumslag-Solitar groups admit EZ-structures.

All generalized Baumslag-Solitar groups admit Z-structures.

Advances understanding of group boundary structures.

Abstract

A $\mathcal{Z}$-structure on a group $G$ was introduced by Bestvina in order to extend the notion of a group boundary beyond the realm of CAT(0) and hyperbolic groups. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $\mathcal{EZ}$-structure. The general questions of which groups admit $\mathcal{Z}$- or $\mathcal{EZ}$-structures remain open. In this paper we add to the current knowledge by showing that all Baumslag-Solitar groups admit $\mathcal{EZ}$-structures and all generalized Baumslag-Solitar groups admit $\mathcal{Z}$-structures.