覆盖R-树、R-自由群和枝晶
Covering R-trees, R-free groups, and dendrites
摘要：
证明了每个长度空间X都是R-树T的轨道空间（具有商度量），通过X的等距的局部自由子群G（X）的自由作用。映射f：T->X是一种广义的覆叠映射，称为URL映射，它在X上的URL映射中是通用的。T是唯一的R树，它允许URL-映上X上。当X是n>1维的完备黎曼流形M，Menger海绵，Sierpin'ski地毯或垫圈，T等距于所谓的“泛”R-树A_{c}，它的价等于连续统在每一点的基数。在这些情况下，当X是夏威夷耳环H时，G（X）对T的作用给出了邓伍迪和扎斯特罗的例子，这些例子否定地回答了J. W. Morgan关于R-树上的群作用。事实上，对于H上的一个长度度量，我们精确地得到了Zastrow的例子。
英文摘要：
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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