On ℝ-trees, homotopies, and coveringmaps

A map p : E 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 -tree T in such a way that the quotient map p : T 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.