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1. 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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2. Rank and Nielsen equivalence in hyperbolic extensions

   https://doi.org/10.1142/S0218196719500176  

   Dowdall, Spencer; Taylor, Samuel J. (June 2019, International Journal of Algebra and Computation)

   In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered [Formula: see text]-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact subgroups of [Formula: see text]. 

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3. Right-angled Artin groups as normal subgroups of mapping class groups

   https://doi.org/10.1112/S0010437X21007417  

   Clay, Matt; Mangahas, Johanna; Margalit, Dan (August 2021, Compositio Mathematica)

   We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $$m$$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara. 

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4. Surface Houghton groups

   https://doi.org/10.1007/s00208-023-02751-2  

   Aramayona, Javier; Bux, Kai-Uwe; Kim, Heejoung; Leininger, Christopher J. (November 2023, Mathematische Annalen)

   Abstract For every$$n\ge 2$$ $n\ge 2$, thesurface Houghton group$${\mathcal {B}}_n$$ $B_n$is defined as the asymptotically rigid mapping class group of a surface with exactlynends, all of them non-planar. The groups$${\mathcal {B}}_n$$ $B_n$are analogous to, and in fact contain, the braided Houghton groups. These groups also arise naturally in topology: every monodromy homeomorphism of a fibered component of a depth-1 foliation of closed 3-manifold is conjugate into some$${\mathcal {B}}_n$$ $B_n$. As countable mapping class groups of infinite type surfaces, the groups$$\mathcal {B}_n$$ $B_n$lie somewhere between classical mapping class groups and big mapping class groups. We initiate the study of surface Houghton groups proving, among other things, that$$\mathcal {B}_n$$ $B_n$is of type$$\text {F}_{n-1}$$ ${\text{F}}_{n-1}$, but not of type$$\text {FP}_{n}$$ ${\text{FP}}_n$, analogous to the braided Houghton groups. 

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5. Largest acylindrical actions and Stability in hierarchically hyperbolic groups

   Abbott, Carolyn; Behrstock, Jason; Durham, Matthew (July 2020, Transactions of the American Mathematical Society)

   We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, right-angled Coxeter groups, most 3–manifold groups, right-angled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a “best” one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasi- convexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and right-angled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. 

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