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1. Cayley Graphs Without a Bounded Eigenbasis

   https://doi.org/10.1093/imrn/rnaa298  

   Sah, Ashwin; Sawhney, Mehtaab; Zhao, Yufei (December 2020, International Mathematics Research Notices)

   Abstract Does every $$n$$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $$O(1/\sqrt{n})$$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $$n$$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $$O(\sqrt{\log n / n})$$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups. 

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2. The Stabilized Automorphism Group of a Subshift

   https://doi.org/10.1093/imrn/rnab204  

   Hartman, Yair; Kra, Bryna; Schmieding, Scott (August 2021, International Mathematics Research Notices)

   Abstract For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group, study its algebraic properties, and use them to distinguish many of the stabilized automorphism groups. We also show that for a full shift, the subgroup of the stabilized automorphism group generated by elements of finite order is simple and that the stabilized automorphism group is an extension of a free abelian group of finite rank by this simple group. 

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3. 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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4. Sato–Tate Groups of Abelian Threefolds

   https://doi.org/10.1090/memo/1582  

   Fité, Francesc; Kedlaya, Kiran; Sutherland, Andrew (August 2025, Memoirs of the American Mathematical Society)

   Given an abelian variety over a number field, its Sato–Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the $L$-function of the abelian variety. It was previously shown by Fité, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato–Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato–Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato–Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura’s theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations inGAP,SageMath, andMagma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated $L$-functions. 

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5. Generically computable Abelian groups

   https://doi.org/10.1007/978-3-031-34034-5_3  

   Calvert, W.; Cenzer, D.; Harizanov, V. (June 2023, Unconventional Computation and Natural Computation)

   Genova, D.; Kari, J. (Ed.)

   Generically computable sets, as introduced by Jockusch and Schupp, have been of great interest in recent years. This idea of approximate computability was motivated by asymptotic density problems studied by Gromov in combinatorial group theory. More recently, we have defined notions of generically computable structures, and studied in particular equivalence structures and injection structures. A structure is said to be generically computable if there is a c.e. substructure defined on an asymptotically dense set, where the functions are computable and the relations are computably enumerable. It turned out that every equivalence structure has a generically computable copy, whereas there is a non-trivial characterization of the injection structures with generically computable copies. In this paper, we return to group theory, as we explore the generic computability of Abelian groups. We show that any Abelian p-group has a generically computable copy and that such a group has a Σ2-generically computably enumerable copy if and only it has a computable copy. We also give a partial characterization of the Σ1-generically computably enumerable Abelian p-groups. We also give a non-trivial characterization of the generically computable Abelian groups that are not p-groups. 

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