# Geometric Analysis and Topology Seminar

#### Combinatorially Loewner Spaces that are not quasisymmetric to Loewner spaces

**Speaker:**
Sylvester Eriksson-Bique, University of Jyväskylä

**Location:**
Online

**Date:**
Friday, October 25, 2024, 11 a.m.

**Synopsis:**

Loewner spaces are metric measure spaces which permit a comprehensive theory of quasisymmetric and quasiconformal maps. If the boundary of a hyperbolic group happens to be quasisymmetric to a Loewner space, then this structure theory can be used to imply Mostow-type rigidity for the group in question. For this reason, we would really like to recognize metric spaces that are quasisymmetric to Loewner spaces. This is a difficult problem, and we can't answer this problem presently even for most explicit fractals (e.g. the Sierpinski carpet). A natural candidate for a characterization is the combinatorial Loewner property, and Kleiner asked in 2006 if all approximately self-similar and combinatorially Loewner metric spaces are quasisymmetric to a Loewner space. In this talk, I present a negative answer to this question and explain an obstruction to being quasisymmetric to a Loewner space. I also show a connection between Kleiners question and another seemingly unrelated question concerning the analytic dimension in the sense of Cheeger. While the counterexamples we find do not seem to arise as group boundaries, this connection appears useful in constructing CLP group boundaries that are not quasisymmetric to Loewner spaces. This is joint work with Guy C. David and Riku Anttila.