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1. Surface groups, infinite generating sets, and stable commutator length

   https://doi.org/10.1017/prm.2019.22  

   Margalit, Dan; Putman, Andrew (April 2019, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   Abstract We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves. 

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2. The coarse geometry of hexagon decomposition graphs

   https://doi.org/10.4153/S0008414X24000853  

   Gültepe, Funda; Parlier, Hugo (January 2025, Canadian Journal of Mathematics)

   We define and study graphs associated to hexagon decompositions of surfaces by curves and arcs. One of the variants is shown to be quasi-isometric to the pants graph, whereas the other variant is quasi-isometric to (a Cayley graph of) the mapping class group. 

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3. Non-planar Ends are Continuously Unforgettable

   https://doi.org/10.1093/imrn/rnaf026  

   Aramayona, Javier; de_Pool, Rodrigo; Skipper, Rachel; Tao, Jing; Vlamis, Nicholas G; Wu, Xiaolei (March 2025, International Mathematics Research Notices)

   Abstract We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups. 

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4. Self-Similar Surfaces: Involutions and Perfection

   https://doi.org/10.1307/mmj/20216114  

   Malestein, Justin; Tao, Jing (July 2024, Michigan Mathematical Journal)

   We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends spaces, as defined by Mann and Rafi, and with 0 or infinite genus, we show 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, we 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, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property. 

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5. Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs

   https://doi.org/10.5802/ahl.238  

   Domat, George; Hoganson, Hannah; Kwak, Sanghoon (January 2025, Annales Henri Lebesgue)

   We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group. We establish a semidirect product decomposition, compute first integral cohomology, and classify when they satisfy residual finiteness and the Tits alternative. These results provide a framework and some initial steps towards quasi-isometric and algebraic rigidity of these groups. 

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