Group boundaries for semidirect products with $\mathbb{Z}$

Abstract

Bestvina's notion of a $\mathcal{Z}$-structure provides a general framework for group boundaries that includes Gromov boundaries of hyperbolic groups and visual boundaries of CAT(0) groups as special cases. A refinement, known as an $E\mathcal{Z}$-structure, has proven useful in attacks on the Novikov conjecture and related problems. Characterizations of groups admitting a $\mathcal{Z}$- or an $E\mathcal{Z}$-structure are longstanding open problems. In this paper, we examine groups of the form $G{⋊}_{\varphi }\mathbb{Z}$. For example, we show that if $G$ is torsion-free and admits a $\mathcal{Z}$-structure, then so does every semidirect product of this type. We prove a similar theorem for $E\mathcal{Z}$-structures, under an additional hypothesis. As applications, we show that all closed 3-manifold groups admit $\mathcal{Z}$-structures, as do all strongly polycyclic groups and all groups of polynomial growth. In those latter cases, our $\mathcal{Z}$-boundaries are always spheres. This allows one to make strong conclusions about the group cohomology and end invariants of those groups. In another direction, we expand upon the notion of an $E\mathcal{Z}$-structure and discuss new applications to the Novikov conjecture.