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arXiv · 0707.3609

Covering R-trees

V. N. Berestovskii·C. Plaut
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Abstract

We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of the fundamental group of X. In particular, the Sierpin'ski gasket and carpet, and the Menger sponge all have the same covering R-tree, which is complete and has at each point valency equal to the continuum. This latter R-tree is of particular interest because it is "universal" in at least two senses: First, every R-tree of valency at most the continuum can be isometrically embedded in it. Second, every Peano continuum is the image of it via an open light mapping. We provide a sketch of our previous construction of the uniform universal cover in the special case of inner metric spaces, the properties of which are used in the proof.

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V. N. Berestovskii, C. Plaut. 2008-12-10. Covering R-trees. https://arxiv.org/abs/0707.3609

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