Trees of metric compacta and trees of manifolds

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.

45 pages

References in corpus (4)

• Trees of manifolds as boundaries of spaces and groups
• Hyperbolic Coxeter groups with Sierpiński carpet boundary
• Gromov boundaries as Markov compacta
• Right-angled Coxeter groups with n-dimensional Sierpiński compacta as boundaries

Cited by in corpus (6)

• Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, with an appendix written jointly with Christopher Cashen
• Connectedness properties and splittings of groups with isolated flats
• Trees of manifolds as boundaries of spaces and groups
• Gromov boundaries as Markov compacta
• Reflection trees of graphs as boundaries of Coxeter groups
• Convex cocompact groups in real hyperbolic spaces with limit set a Pontryagin sphere