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1. Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

   https://doi.org/10.1007/s00209-022-03193-3  

   Larsen, Michael; Taylor, Jay; Tiep, Pham Huu (February 2023, Mathematische Zeitschrift)

   Abstract For every integer k there exists a bound $$B=B(k)$$ B = B ( k ) such that if the characteristic polynomial of $$g\in \textrm{SL}_n(q)$$ g ∈ SL n ( q ) is the product of $$\le k$$ ≤ k pairwise distinct monic irreducible polynomials over $$\mathbb {F}_q$$ F q , then every element x of $$\textrm{SL}_n(q)$$ SL n ( q ) of support at least B is the product of two conjugates of g . We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions ( p ,  q ), in the special case that $$n=p$$ n = p is prime, if g has order $$\frac{q^p-1}{q-1}$$ q p - 1 q - 1 , then every non-scalar element $$x \in \textrm{SL}_p(q)$$ x ∈ SL p ( q ) is the product of two conjugates of g . The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups. 

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2. Invariant random subgroups of semidirect products

   https://doi.org/10.1017/etds.2018.46  

   BIRINGER, IAN; BOWEN, LEWIS; TAMUZ, OMER (January 2018, Ergodic Theory and Dynamical Systems)

   We study invariant random subgroups (IRSs) of semidirect products $$G=A\rtimes \unicode[STIX]{x1D6E4}$$ . In particular, we characterize all IRSs of parabolic subgroups of $$\text{SL}_{d}(\mathbb{R})$$ , and show that all ergodic IRSs of $$\mathbb{R}^{d}\rtimes \text{SL}_{d}(\mathbb{R})$$ are either of the form $$\mathbb{R}^{d}\rtimes K$$ for some IRS of $$\text{SL}_{d}(\mathbb{R})$$ , or are induced from IRSs of $$\unicode[STIX]{x1D6EC}\rtimes \text{SL}(\unicode[STIX]{x1D6EC})$$ , where $$\unicode[STIX]{x1D6EC}<\mathbb{R}^{d}$$ is a lattice. 

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3. 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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4. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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5. Polynomial Parametrization for SL2 over Quadratic Number Rings

   https://doi.org/10.1093/imrn/rnz046  

   Larsen, Michael; Nguyen, Dong Quan (March 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract If $$R$$ is the ring of integers of a number field, then there exists a polynomial parametrization of the set $$\operatorname{SL}_2(R)$$, that is, an element $$A\in{\textrm{SL}}_2(\mathbb{Z}[x_1,\ldots ,x_n])$$ such that every element of $$\operatorname{SL}_2(R)$$ is obtained by specializing $$A$$ via some homomorphism $$\mathbb{Z}[x_1,\ldots ,x_n]\to R$$. 

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