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The class of self-similar 2-manifolds consists of manifolds exhibiting a typeof homogeneity akin to the 2-sphere and the Cantor set, and includes both the2-sphere and the 2-sphere with a Cantor set removed. This chapter aims toprovide a narrative thread between recent results on the structure ofhomeomorphism groups/mapping class groups of self-similar 2-manifolds, and alsoconnections to classical structural results on the homeomorphism group of the2-sphere and the Cantor set. In order to do this, we provide a survey of recentresults, an exposition on classical results about homeomorphism groups, providea treatment of the structure of stable sets, and proveextensions/strengthenings of the recent results surveyed.  Of particular note, we establish the following theorems: (1) Acharacterization of homeomorphisms of (orientable) perfectly self-similar2-manifolds that normally generate the group of (orientation-preserving)homeomorphisms -- a strengthening of a result of Malestein--Tao. (2) Thehomeomorphism group of a perfectly self-similar 2-manifold is stronglydistorted -- an extension of a result of Calegari--Freedman for spheres. (3)The homeomorphism group of a perfectly tame 2-manifold is Steinhaus, and hencehas the automatic continuity property -- an extension of a result of Mann indimension two -- providing the first examples of homeomorphism groups ofinfinite-genus 2-manifolds with the Steinhaus property.

Homeomorphism groups of self-similar 2-manifolds