Group boundaries for semidirect products with Z

TL;DR

This paper investigates how group boundaries, specifically Z-structures and EZ-structures, behave under semidirect products with Z, providing new results for various classes of groups and applications to the Novikov Conjecture.

Contribution

It proves that semidirect products of torsion-free groups with Z preserve Z-structures and extends EZ-structure applicability, with implications for group cohomology and the Novikov Conjecture.

Findings

All closed 3-manifold groups admit Z-structures.

Strongly polycyclic groups have Z-structures with sphere boundaries.

Groups of polynomial growth have Z-boundaries as spheres.

Abstract

Bestvina's notion of a 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 EZ-structure has proven useful in attacks on the Novikov Conjecture and related problems. Characterizations of groups admitting a Z- or EZ-structure are longstanding open problems. In this paper, we examine semidirect products of a group G with the integers. For example, we show that, if G is torsion-free and admits a Z-structure, then so does every semidirect product of this type. We prove a similar theorem for EZ-structures, under an additional hypothesis. As applications, we show that all closed 3-manifold groups admit Z-structures, as do all strongly polycyclic groups, and all groups of polynomial growth. In those latter two cases our 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 EZ-structure and discuss new applications to the Novikov Conjecture.