Homeomorphism Groups of Self-Similar 2-Manifolds

Nicholas G. Vlamis ()
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Nicholas G. Vlamis: CUNY Graduate Center, Department of Mathematics

Chapter Chapter 5 in In the Tradition of Thurston III, 2024, pp 105-167 from Springer

Abstract: 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.

Keywords: Big mapping class group; Infinite-type surface; Self-similar 2-manifold; Strong distortion; Automatic continuity; Rokhlin property; Coarsely bounded groups; 57K20; 57S05; 54H11; 20F65 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/978-3-031-43502-7_5

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