Given a finite group $G$, we prove that there exist signs $\varepsilon\in\{\pm1\}^G$ such that $$\left\| \sum_{g\in G} \varepsilon_g\rho(g) \right\|\leq C\, \sqrt{|G|},$$ where $\rho$ is the left regular representation of $G$, and $C$ is a universal constant. This special case of the Matrix Spencer conjecture was posed in [BKMZ24], where it was established for simple groups.

We study the full automorphism groups of smooth cubic fourfolds with prescribed symplectic automorphism group. Our starting point is the classification of symplectic automorphism groups by Laza and Zheng. We focus on the non-symplectic index, namely, the index of the symplectic automorphism group in the full automorphism group. We prove general restrictions on this index. We also compute bounds by group-theoretic and lattice-theoretic methods. In several cases, we determine all possible indices. For coinvariant lattices of rank 19, we classify all possible pairs consisting of the symplectic automorphism group and the full automorphism group.

We develop a new technique to upgrade relative biexactness in general von Neumann algebras: suppose that $\{N_i\}_{i\in I}\subset M$ are mixing and biexact subalgebras of a separable von Neumann algebra with expectation, and if $M$ is biexact relative to $\{N_i\}_{i\in I}$, then $M$ is biexact. This result yields several new examples of biexact von Neumann algebras, notably including amalgamated free products. By generalizing the relative biexactness results of Hoshino to the von Neumann algebra setting and applying our result above along with certain bimodule computations, we in fact obtain, as an application, a new classification result for biexactness for graph products of finite dimensional von Neumann algebras. This yields significant extensions of prior works of Caspers-Borst, and Blufstein-Goldman-Oyakawa.

We give a dynamical proof of Benoist's non-arithmeticity theorem for Jordan spectra of Zariski dense subgroups of connected semisimple real algebraic groups. After passing to a Zariski dense Schottky subgroup, we use the coding of the limit set to realize Jordan projections as periods of a vector-valued Busemann return map for an expanding map on the Furstenberg boundary. The key step is to prove that a suitable two-branch asymptotic discrepancy is not locally constant on the limit set. We also show that the same criterion applies beyond Lie groups; in particular, it yields a direct density result for multiplier spectra of hyperbolic rational maps whose Julia set is not contained in a circle.

Let $G$ be the group of power series $x+a_2x^2+a_3x^3+\cdots\in R[[x]]$ under substitution, where $R$ is a commutative ring with $1\neq 0$ of prime characteristic $p$. Given any $n\geq 1$, the subgroup $K_n=\{x+a_{n+1}x^{n+1}+a_{n+2}x^{n+2}+\cdots\,|\, a_i\in R\}$ is normal in $G$, and the quotient $G_n=G/K_n$ is the group of truncated polynomials over $R$ of degree $\leq n$ under substitution. In this paper, we compute the exponent of the image of $K_r$ in $G_n$, for all $r,n\geq 1$, indicating in every case a family of elements realizing this exponent.

For primes $p\geq 5$, we determine those $q$ for which the projective special linear group $\psl{2}{q}$ over the finite field of order $q$ is $(2,p)$-generated -- that is, there exist two elements of $\psl{2}{q}$ of orders $2$ and $p$, respectively, that generate $\psl{2}{q}$.

It is a study note detailing the connection between Boolean inverse monoids and ample groupoids.

We prove that groups of the form $\mathbb Z^m {\,\rm wr\,} \mathbb Z^n$, where $m,n \in \mathbb N$, are regularly bi-interpretable with $\mathbb Z$ and therefore are first-order rigid: every finitely generated group elementarily equivalent to $\mathbb Z^m {\,\rm wr\,} \mathbb Z^n$ is isomorphic to $\mathbb Z^m {\,\rm wr\,} \mathbb Z^n$. On the other hand, we show that $\mathbb Z^2 {\,\rm wr\,} \mathbb Z$ admits $2^{\aleph_0}$ elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.

In our previous paper, we gave a complete list of the finite non-abelian simple groups whose holomorph contains a solvable regular subgroup. In this paper, we refine our previous work by considering all finite almost simple groups. In particular, our result yields a complete characterization of the finite almost simple groups which occur as the type of a Hopf-Galois structure on a solvable extension, or equivalently, the additive group of a skew brace having a solvable multiplicative group.