The dense amalgam of metric compacta and topological characterization of boundaries of free products of groups

TL;DR

This paper introduces the dense amalgam operation for compact metric spaces and demonstrates its use in characterizing the boundaries of free product groups and Coxeter groups, providing new topological insights.

Contribution

The paper defines the dense amalgam operation and applies it to describe the ideal boundaries of free products and Coxeter groups, extending topological characterizations.

Findings

Boundaries of free products are homeomorphic to dense amalgams of factor boundaries.

Boundaries of certain Coxeter groups are dense amalgams of their subgroups' boundaries.

Provides a characterization of dense amalgams via specific properties.

Abstract

We introduce and study the operation, called dense amalgam, which to any tuple X_1,...,X_k of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted $\widetilde\sqcup(X_1,...,X_k)$, in which there are many appropriately distributed copies of the spaces X_1,...,X_k. We then show that, in various settings, the ideal boundary of the free product of groups is homeomorphic to the dense amalgam of boundaries of the factors. We give also related more general results for graphs of groups with finite edge groups. We justify these results by referring to a convenient characterization of dense amalgams, in terms of a list of properties, which we also provide in the paper. As another application, we show that the boundary of a Coxeter group which has infinitely many ends, and which is not virtually free, is the dense amalgam of the boundaries of its maximal 1-ended special subgroups.