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1. Flatness of the Commutator Map Over SLn

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

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

   null (Ed.)

   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. Characterizing Schwarz maps by tracial inequalities

   https://doi.org/10.1007/s11005-023-01636-4  

   Carlen, Eric; Müller-Hermes, Alexander (February 2023, Letters in Mathematical Physics)

   Abstract Let $$\phi $$ ϕ be a positive map from the $$n\times n$$ n × n matrices $$\mathcal {M}_n$$ M n to the $$m\times m$$ m × m matrices $$\mathcal {M}_m$$ M m . It is known that $$\phi $$ ϕ is 2-positive if and only if for all $$K\in \mathcal {M}_n$$ K ∈ M n and all strictly positive $$X\in \mathcal {M}_n$$ X ∈ M n , $$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$ ϕ ( K ∗ X - 1 K ) ⩾ ϕ ( K ) ∗ ϕ ( X ) - 1 ϕ ( K ) . This inequality is not generally true if $$\phi $$ ϕ is merely a Schwarz map. We show that the corresponding tracial inequality $${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$ Tr [ ϕ ( K ∗ X - 1 K ) ] ⩾ Tr [ ϕ ( K ) ∗ ϕ ( X ) - 1 ϕ ( K ) ] holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results. 

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3. 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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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. 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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