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1. The inductive McKay–Navarro conditions for the prime 2 and some groups of Lie type

   https://doi.org/10.1090/bproc/123  

   Ruhstorfer, L.; Schaeffer Fry, A. (April 2022, Proceedings of the American Mathematical Society, Series B)

   For a prime ℓ \ell , the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with ℓ ′ \ell ’ -degree and the corresponding set for the normalizer of a Sylow ℓ \ell -subgroup. Navarro’s refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro–Späth–Vallejo have reduced the McKay–Navarro conjecture to certain “inductive” conditions on finite simple groups. We prove that these inductive McKay–Navarro (also called the inductive Galois–McKay) conditions hold for the prime ℓ = 2 \ell =2 for several groups of Lie type, namely the untwisted groups without non-trivial graph automorphisms. 

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2. Profinite properties of algebraically clean graphs of free groups

   Jankiewicz, Kasia; Schreve, Kevin (January 2025, Journal of pure and applied algebra)

   We prove that for every prime p algebraically clean graphs of groups are vir- tually residually p-finite and cohomologically p-complete. We also prove that they are cohomologically good. We apply this to certain 2-dimensional Artin groups. 

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3. Critical Groups of Arithmetical Structures on Star Graphs and Complete Graphs

   https://doi.org/10.37236/11729  

   Archer, Kassie; Diaz-Lopez, Alexander; Glass, Darren Broderick; Louwsma, Joel (January 2024, The Electronic Journal of Combinatorics)

   An arithmetical structure on a finite, connected graph without loops is an assignment of positive integers to the vertices that satisfies certain conditions. Associated to each of these is a finite abelian group known as its critical group. We show how to determine the critical group of an arithmetical structure on a star graph or complete graph in terms of the entries of the arithmetical structure. We use this to investigate which finite abelian groups can occur as critical groups of arithmetical structures on these graphs. 

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4. Finite quotients of 3-manifold groups

   https://doi.org/10.1007/s00222-024-01262-4  

   Sawin, Will; Wood, Melanie Matchett (July 2024, Inventiones mathematicae)

   For and finite groups, does there exist a 3-manifold group with as a quotient but no as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments. 

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5. Realizing finite groups as automizers

   https://doi.org/10.1515/jgth-2022-0145  

   Bayard, Sylvia; Lynd, Justin (February 2024, Journal of Group Theory)

   Abstract It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺.The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park. 

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