Homeomorphism groups of self-similar 2-manifolds

Abstract

The class of self-similar 2-manifolds consists of manifolds exhibiting a type of homogeneity akin to the 2-sphere and the Cantor set, and includes both the 2-sphere and the 2-sphere with a Cantor set removed. This chapter aims to provide a narrative thread between recent results on the structure of homeomorphism groups/mapping class groups of self-similar 2-manifolds, and also connections to classical structural results on the homeomorphism group of the 2-sphere and the Cantor set. In order to do this, we provide a survey of recent results, an exposition on classical results about homeomorphism groups, provide a treatment of the structure of stable sets, and prove extensions/strengthenings of the recent results surveyed. Of particular note, we establish the following theorems: (1) A characterization of homeomorphisms of (orientable) perfectly self-similar 2-manifolds that normally generate the group of (orientation-preserving) homeomorphisms -- a strengthening of a result of Malestein-Tao. (2) The homeomorphism group of a perfectly self-similar 2-manifold is strongly distorted -- an extension of a result of Calegari-Freedman for spheres. (3) The homeomorphism group of a perfectly tame 2-manifold is Steinhaus, and hence has the automatic continuity property -- an extension of a result of Mann in dimension two -- providing the first examples of homeomorphism groups of infinite-genus 2-manifolds with the Steinhaus property.

Figures (8)

• Figure 1: The setup for one side of the double slide $s_{b,c}$, and hence the setup for a boundary slide. The disk with the "X" represents a crosscap, meaning the open disk is removed and the antipodal points of the boundary circle are identified.
• Figure 2: The support of a $D$-translation sending $D_n$ to $D_{n+1}$ for all $n \in \mathbb Z$, as given by Lemma \ref{['lem:translation']}.
• Figure 3: A schematic of the subsurfaces in the proof of Lemma \ref{['lem:factor2']}.
• Figure 4: A schematic of the sets in the Proof of Theorem \ref{['thm:cb']} with $M = \mathbb{R}^2$.
• Figure 5: A schematic of the subsurfaces in the proof of Lemma \ref{['lem:dense2']} with $M = \mathbb{R}^2$.

Theorems & Definitions (144)

• Definition 1.1: Self-similar 2-manifold
• Corollary 2.1: Anderson AndersonAlgebraic
• Corollary 2.2
• Corollary 2.3
• Corollary 2.4: Purity Theorem
• Corollary 2.5
• Corollary 2.6
• Corollary 2.7: Calegari--Freedman CalegariDistortion
• Corollary 2.8
• Corollary 2.9