From Cuts to R trees

TL;DR

This paper establishes precise conditions for constructing a canonical R-tree from a collection of separators in a connected topological space, ensuring group actions extend naturally to the tree.

Contribution

It introduces sharp conditions for creating R-trees from separators in topological spaces and demonstrates how group actions extend to these trees.

Findings

Conditions for R-tree construction are precisely characterized.

Group actions on the topological space extend to the R-tree.

The construction provides a canonical form for analyzing topological group actions.

Abstract

We provide sharp conditions under which a collection of separators A of a connected topological space Z leads to a canonical R-tree T . Any group acting on Z by homeomorphisms will act by homeomorphisms on T.