Covering R-trees, R-free groups, and dendrites

Abstract

We prove that every length space X is the orbit space (with the quotient metric) of an R-tree X via a free action of a locally free subgroup Γ (X) of isometries of X. The mapping φ : X → X is a kind of generalized covering map called a URL-map and is universal among URL-maps onto X. X is the unique R-tree admitting a URL-map onto X. When X is a complete Riemannian manifold M n of dimension n 2, the Menger sponge, the Sierpin'ski carpet or gasket, X is isometric to the so-called "universal" R-tree A c , which has valency c = 2 ℵ 0 at each point. In these cases, and when X is the Hawaiian earring H , the action of Γ (X) on X gives examples in addition to those of Dunwoody and Zastrow that negatively answer a question of J.W. Morgan about group actions on R-trees. Indeed, for one length metric on H , we obtain precisely Zastrow's example.