On the dimension distortions of quasi-symmetric homeomorphisms

TL;DR

This paper investigates how quasi-symmetric homeomorphisms distort dimensions, generalizing previous results for Fuchsian groups and showing limitations for certain covering groups, revealing complex geometric behaviors.

Contribution

It extends dimension distortion results to divergence type non-lattice Fuchsian groups and demonstrates the failure of these results for higher-dimensional jungle gym groups.

Findings

Max(dimE, dimh(R\E))=1 for certain Fuchsian groups

Dimension distortion results do not hold for all 'd-dimensional jungle gym' groups

Generalizes previous work on 1-dimensional jungle gyms to higher dimensions

Abstract

In this paper, we first generalize a result of Bishop and Steger [Representation theoretic rigidity in PSL(2, R). Acta Math., 170, (1993), 121-149] by proving that for a Fuchsian group $G$ of divergence type and non-lattice, if $h$ is a quasi-symmetric homeomorphism of the real axis $\mathbb{R}$ corresponding to a quasi-conformal compact deformation of $G$. Then for any $E\subset \mathbb{R}$, we have max(dim$E$, dim$h(\mathbb{R}\setminus E))=1$. Furthermore, we showed that Bishop and steger's result does not hold for the covering groups of all '$d$-dimensional jungle gym' (d is any positive integer) which generalizes G\"onye's results [ Differentiability of quasi-conformal maps on the jungle gym. Trans. Amer. Math. Soc. Vol 359 (2007), 9-32] where the author discussed the case of '$1$-dimensional jungle gym'.