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

TL;DR

This paper establishes a universal covering theory for length spaces using R-trees, introducing URL-maps, and explores specific cases including manifolds, fractals, and the Hawaiian earring, with implications for group actions on R-trees.

Contribution

It introduces the concept of URL-maps as generalized coverings for length spaces and characterizes the universal R-tree covering, extending group action examples to new spaces.

Findings

Every length space is an orbit space of an R-tree via a free isometric action.

The universal R-tree for certain spaces has valency of continuum at each point.

Provides new examples of group actions on R-trees, answering open questions.

Abstract

We prove that every length space X is the orbit space (with the quotient metric) of an R-tree T via a free action of a locally free subgroup G(X) of isometries of X. The mapping f:T->X is a kind of generalized covering map called a URL-map and is universal among URL-maps onto X. T is the unique R-tree admitting a URL-map onto X. When X is a complete Riemannian manifold M of dimension n>1, the Menger sponge, the Sierpin'ski carpet or gasket, T is isometric to the so-called "universal" R-tree A_{c}, which has valency equal to the cardinality of the continuum at each point. In these cases, and when X is the Hawaiian earring H, the action of G(X) on T 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.