Analytically one-dimensional planes and the Combinatorial Loewner Property

TL;DR

This paper explores the conditions under which metric spaces are quasisymmetric to Loewner spaces, constructing examples that challenge existing conjectures and potentially resolve open questions about analytically 1-dimensional planes.

Contribution

It constructs examples that either serve as new counterexamples to Kleiner's conjecture or resolve the existence of analytically 1-dimensional planes, advancing understanding in metric space analysis.

Findings

Constructed examples challenge Kleiner's conjecture.

Showed the possibility of analytically 1-dimensional planes in certain metric spaces.

Provided insights into the structure of quasisymmetric spaces and their analytic dimensions.

Abstract

It is a major problem in analysis on metric spaces to understand when a metric space is quasisymmetric to a space with strong analytic structure, a so-called Loewner space. A conjecture of Kleiner, recently disproven by Anttila and the second author, proposes a combinatorial sufficient condition. The counterexamples constructed are all topologically one dimensional, and the sufficiency of Kleiner's condition remains open for most other examples. A separate question of Kleiner and Schioppa, apparently unrelated to the problem above, asks about the existence of "analytically $1$-dimensional planes": metric measure spaces quasisymmetric to the Euclidean plane but supporting a $1$-dimensional analytic structure in the sense of Cheeger. In this paper, we construct an example for which the conclusion of Kleiner's conjecture is not known to hold. We show that either this conclusion fails in our example or there exists an "analytically $1$-dimensional plane". Thus, our construction either yields a new counterexample to Kleiner's conjecture, different in kind from those of Anttila and the second author, or a resolution to the problem of Kleiner--Schioppa.