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

Abstract

We introduce and study the operation, called dense amalgam, which to any tuple $X_1,\dots ,X_k$ of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted $\tilde{\bigsqcup }(X_1,\dots ,X_k)$, in which there are many appropriately distributed copies of the spaces $X_1,\dots ,X_k$. 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.