2009/04/23 by V. N. Berestovskii, Berestovskii, V. N., C. P. Plaut +1
Mathematics · #20F65 #28A80 #53C23 #54F15 #54F50 #57M07 #FOS: Mathematics #General Topology (math.GN) #Metric Geometry (math.MG) #math.GN #math.MG #msc:20F65 #msc:28A80 #msc:53C23 #msc:54F15 #msc:54F50 #msc:57M07
paper · pdf · doi:10.48550/arxiv.0904.3767
This paper is the result of splitting off some of the results in the preprint "Covering R-trees" and adding additional applications to R-free groups
arxiv created 2009/04/23 · arxiv updated 2009/12/01
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 Ac, 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.