Self-similar surfaces: involutions and perfection (Q6590153)

Let \(\Sigma\) be a connected and oriented surface and let \(G(\Sigma)\) be either the group \(\mathrm{Homeo}^{+}(\Sigma)\) of orientation preserving self-homeomorphisms of \(\Sigma\) or the mapping class group \(\mathrm{MCG}(\Sigma)\) of \(\Sigma\). When \(\Sigma\) has finite type, this topology is discrete and \(\mathrm{MCG}(\Sigma)\) is finitely presented. When \(\Sigma\) has infinite type, then \(\mathrm{MCG}(\Sigma)\) is also a non-locally-compact Polish group, similar to the homeomorphism group (in particular, \(\mathrm{MCG}(\Sigma)\) is not countably generated, justifying the nomenclature of big mapping class group).\N\NThe purpose of this paper is to investigate the problem of when big mapping class groups are generated by involutions. The authors restrict their attention to the class of self-similar surfaces, which are surfaces with self-similar ends spaces, and with \(0\) or infinite genus (as defined by \textit{K. Mann} and \textit{K. Rafi} [Geom. Topol. 27, No. 6, 2237--2296 (2023; Zbl 1545.57011)]). They prove that when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, they derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, they produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements.