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1. Flatness of the Commutator Map Over $$\textrm{SL}_n$$

   https://doi.org/10.1093/imrn/rnz285  

   Larsen, Michael; Lu, Zhipeng (November 2019, International Mathematics Research Notices)

   Abstract Let $$K$$ be any field, and let $$n$$ be a positive integer. If we denote by $$\xi _{\textrm{SL}_n}\colon \textrm{SL}_n\times \textrm{SL}_n\to \textrm{SL}_n$$ the commutator morphism over $$K$$, then $$\xi _{\textrm{SL}_n}$$ is flat over the complement of the center of $$\textrm{SL}_n$$. 

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2. Zero Cycles on a Product of Elliptic Curves Over a p -adic Field

   https://doi.org/10.1093/imrn/rnab020  

   Gazaki, Evangelia; Leal, Isabel (March 2021, International Mathematics Research Notices)

   Abstract We consider a product $$X=E_1\times \cdots \times E_d$$ of elliptic curves over a finite extension $$K$$ of $${\mathbb{Q}}_p$$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of $$X$$ is the direct sum of a finite group and a divisible group, extending work by Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $$CH_0(X)/p^n\rightarrow H^{2d}_{\acute{\textrm{e}}\textrm{t}}(X, \mu _{p^n}^{\otimes d})$$. We give specific criteria that guarantee this map is injective for every $$n\geq 1$$. When all curves have good ordinary reduction, we show that it suffices to extend to a specific finite extension $$L$$ of $$K$$ for these criteria to be satisfied. This extends previous work by Yamazaki and Hiranouchi. 

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3. 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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4. Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

   https://doi.org/10.1007/s11856-024-2613-1  

   Hung, Nguyen Ngoc; Sambale, Benjamin; Tiep, Pham Huu (September 2024, Israel Journal of Mathematics)

   Abstract Letk(B₀) andl(B₀) respectively denote the number of ordinary andp-Brauer irreducible characters in the principal blockB₀of a finite groupG. We prove that, ifk(B₀)−l(B₀) = 1, thenl(B₀) ≥p− 1 or elsep= 11 andl(B₀) = 9. This follows from a more general result that for every finite groupGin which all non-trivialp-elements are conjugate,l(B₀) ≥p− 1 or elsep= 11 and$$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ $G/O_{p^{\prime }}\left(G\right)\cong C_{11}^2⋊\text{SL}(2,5)$. These results are useful in the study of principal blocks with few characters. We propose that, in every finite groupGof order divisible byp, the number of irreducible Brauer characters in the principalp-block ofGis always at least$$2\sqrt {p - 1} + 1 - {k_p}(G)$$ $2\sqrt{p-1}+1-k_p\left(G\right)$, wherek_p(G) is the number of conjugacy classes ofp-elements ofG. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number ofp-regular classes in finite groups. 

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5. Determinantal Representations and the Image of the Principal Minor Map

   https://doi.org/10.1093/imrn/rnae038  

   Al_Ahmadieh, Abeer; Vinzant, Cynthia (March 2024, International Mathematics Research Notices)

   Abstract In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $$(\textrm{SL}_{2}(\mathbb{R}))^{n} \rtimes S_{n}$$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $$\mathbb{F}$$, there is no finite set of equations whose orbit under $$(\textrm{SL}_{2}(\mathbb{F}))^{n} \rtimes S_{n}$$ cuts out the image of $$n\times n$$ matrices over $${\mathbb{F}}$$ under the principal minor map for every $$n$$. 

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