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

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.