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1. Abelian varieties with isogenous reductions

   https://doi.org/https://doi.org/10.5802/crmath.129  

   Khare, Chandreshekhar; Larsen, Michael: (October 2020, Comptes rendus)

   null (Ed.)

   Let $$A_1$$ and $$A_2$$ be abelian varieties over a number field $$K$$. We prove that if there exists a non-trivial morphism of abelian varieties between reductions of $$A_1$$ and $$A_2$$ at a sufficiently high percentage of primes, then there exists a non-trivial morphism $$A_1\to A_2$$ over $$\bar K$$. Along the way, we give an upper bound for the number of components of a reductive subgroup of $$GL_n$$ whose intersection with the union of $$Q$$-rational conjugacy classes of $$GL_n$$ is Zariski-dense. This can be regarded as a generalization of the Minkowski-Schur theorem on faithful representations of finite groups with rational characters. 

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2. CHARACTER LEVELS AND CHARACTER BOUNDS

   https://doi.org/10.1017/fmp.2019.9  

   Guralnick, Robert; Larsen, Michael; Tiep, Pham Huu (January 2020, Forum of mathematics)

   We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’] 

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3. Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

   https://doi.org/10.1353/ajm.2024.a928325  

   Kuhn, Nicholas J; Lloyd, Christopher_J R (June 2024, American Journal of Mathematics)

   abstract: In the early 1940s, P. A. Smith showed that if a finite $$p$$-group $$G$$ acts on a finite dimensional complex $$X$$ that is mod $$p$$ acyclic, then its space of fixed points, $X^G$, will also be mod $$p$$ acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $$G$$-space $$X$$ is a equivariant homotopy retract of the $$p$$-localization of a based finite $$G$$-C.W. complex, given $H 

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4. Finite image homomorphisms of the braid group and its generalizations

   https://doi.org/10.1017/S0017089523000022  

   Scherich, Nancy; Verberne, Yvon (March 2023, Glasgow Mathematical Journal)

   Abstract. Using totally symmetric sets, Chudnovsky–Kordek–Li–Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups. 

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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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