Combinatorial R-trees as generalized Cayley graphs for fundamental groups of one-dimensional spaces

TL;DR

This paper introduces a novel combinatorial structure called combinatorial R-trees, which generalize Cayley graphs to represent fundamental groups of one-dimensional spaces, addressing a question posed by Cannon and Conner.

Contribution

It provides the first combinatorial model of topological Cayley graphs for fundamental groups of one-dimensional spaces, expanding the understanding of their structure.

Findings

Defines combinatorial R-trees as generalized Cayley graphs

Establishes a correspondence between R-trees and fundamental groups

Answers Cannon and Conner's question positively

Abstract

In their study of fundamental groups of one-dimensional path-connected compact metric spaces, Cannon and Conner have asked: Is there a tree-like object that might be considered the topological Cayley graph? We answer this question in the positive and provide a combinatorial description of such an object.