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

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

2009-04-23

arxiv.org/abs/0904.3767v1

Mathematics

Metric Geometry

This paper is the result of splitting off some of the results in the preprint "Covering R-trees" and adding additional applica

Scientific paper

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.

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