Covering R-trees

TL;DR

This paper introduces the concept of covering R-trees for inner metric spaces, showing they serve as universal metric quotients with applications to fractals and Peano continua, revealing new structural insights.

Contribution

It establishes that every inner metric space has a covering R-tree that acts as a universal metric quotient, extending the understanding of space coverings and embeddings.

Findings

Every inner metric space is a quotient of a complete R-tree.

The covering R-tree is universal for spaces with valency up to the continuum.

Fractals like the Sierpin'ski gasket share the same covering R-tree.

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.