On $\mathbb{R}$-trees, homotopies, and covering maps

Jeremy Brazas, Gregory R. Conner, Paul Fabel and Curtis Kent

Vol. 343 (2026), No. 2, 315–341

DOI: 10.2140/pjm.2026.343.315

Abstract

A map $p:E\to X$ has the unique path lifting property if every path in $X$, after a choice of an initial point, lifts uniquely to a path in $E$. We prove that if a group $G$ acts on an $ℝ$-tree $T$ in such a way that the quotient map $p:T\to T∕G$ has the unique path lifting property, then the quotient space $T∕G$ does not contain a disc. As a consequence, we show that every map of manifolds with the unique path lifting property is a covering map. The proof requires a study of one-dimensional backtracking in paths. We show the surprising and counterintuitive result that the equivalence relation given by homotopies of paths rel. endpoints is generated by inserting and deleting one-dimensional backtracking.

Keywords

$\mathbb{R}$-tree, geodesic $\mathbb{R}$-tree reduction, path homotopy, dendrite, unique path lifting, covering map

Mathematical Subject Classification

Primary: 54F50, 55R65, 57M10

Milestones

Received: 4 March 2025

Revised: 7 April 2026

Accepted: 13 April 2026

Published: 22 May 2026