Trees of manifolds as boundaries of spaces and groups

TL;DR

This paper demonstrates that trees of manifolds serve as boundaries at infinity for various hyperbolic groups, Coxeter groups, and fundamental groups of singular spaces, linking topological and geometric group theory.

Contribution

It establishes new instances of trees of manifolds as Gromov boundaries for hyperbolic and Coxeter groups, expanding understanding of boundaries in geometric group theory.

Findings

Trees of manifolds appear as Gromov boundaries of hyperbolic groups.

They also serve as boundaries of Coxeter groups with manifold nerves.

Boundaries of singular spaces from hyperbolic manifolds are also trees of manifolds.

Abstract

We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary dimension, obtained by the procedure of strict hyperbolization. We also recognize these spaces as boundaries of arbitrary Coxeter groups with manifold nerves, and as Gromov boundaries of the fundamental groups of singular spaces obtained from some finite volume hyperbolic manifolds by cutting off their cusps and collapsing the resulting boundary tori to points.