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

AbstractWe 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 Mn of dimension n⩾2, the Menger sponge, the Sierpin'ski carpet or gasket, X¯ is isometric to the so-called “universal”R-tree Ac, which has valency c=2ℵ0 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