Homeomorphism Groups of Self-Similar 2-Manifolds

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.