$E\mathcal{Z}$-boundaries, splittings over finite subgroups, and dense amalgams

TL;DR

This paper explores the structure of boundaries of infinitely ended groups within the $E\mathcal{Z}$-boundary framework, revealing they can be described as dense amalgams of limit sets of subgroup splittings, unifying various boundary theories.

Contribution

It establishes that boundaries of infinitely ended groups with finite subgroup splittings are dense amalgams of subgroup limit sets within the $E\mathcal{Z}$-boundary framework, unifying multiple boundary concepts.

Findings

Boundaries of infinitely ended groups can be expressed as dense amalgams.

The framework unifies Gromov, CAT(0), and systolic boundaries.

Provides a canonical construction for boundaries in this setting.

Abstract

The dense amalgam is an operation (introduced in arXiv:1410.4989) which to any finite collection of metrizable compacta associates canonically some new highly disconnected compact metrisable space in which embedded copies of the initial spaces are appropriately uniformly and disjointly distributed. We show that in the very general framework of $E\mathcal{Z}$-boundaries (unifying many frameworks such as Gromov boundaries, CAT(0)-boundaries, systolic boundaries, etc.), any boundary $Z$ of an infinitely ended group $\Gamma$ equipped with an appropriate splitting along finite subgroups has a form of the dense amalgam of the limit sets in $Z$ of the factor subgroups of this splitting.