Trees of metric compacta and trees of manifolds

TL;DR

This paper introduces a general construction called the limit of a tree system of spaces, producing complex compact metric spaces from regular spaces like manifolds, with potential applications to the boundaries of infinite groups.

Contribution

It presents a more general and flexible approach to constructing trees of spaces, extending and refining previous results on trees of manifolds and related spaces.

Findings

Clarifies and extends properties of trees of manifolds

Provides a more general framework for tree of spaces construction

Potentially characterizes boundaries of infinite groups

Abstract

We present a construction, called the limit of a tree system of spaces (or, less formally, a tree of spaces). The construction is designed to produce compact metric spaces that resemble fractals, out of more regular spaces, such as closed manifolds, compact polyhedra, compact Menger manifolds, etc. Such spaces are potential candidates to be homeomorphic to ideal boundaries of infinite groups. A very special case of this construction, trees of manifolds (known also as Jakobsche spaces), has been studied in the literature. We present here a different approach, much more general, and, as we believe, much more convenient for establishing various basic properties of the resulting spaces, in a more general setting. Already in the case of trees of manifolds, using this approach we clarify, correct and extend so far known results and properties.