Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb N$ (the non-negative integers under addition). Denote by $\mathcal P_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set. Tringali and Yan [JCTA, 2025] have recently established that $\mathcal P_{\text{fin},0}(\mathbb N)$ has a unique non-trivial automorphism, and conjectured that the automorphism group of $\mathcal P_{\text{fin},0}(H)$ is trivial whenever $H \ne \mathbb N$. We prove this conjecture and, as a byproduct, give a new proof of the Tringali--Yan theorem.
Let $\widetilde{S}\to S$ be an unbranched regular $p$-fold cyclic cover of a closed orientable surface $S$ of genus $g$. Two natural groups are associated with this cover. The first is the centralizer in $\mathrm{Mod}(\widetilde{S})$ of a chosen generator $\sigma$ of the deck transformation group, denoted by $\mathrm{Mod}(\widetilde{S},\sigma)$. The second is the finite-index subgroup of $\mathrm{Mod}(S)$ consisting of mapping classes that fix the nonzero class $[\beta]\in H_1(S;\mathbb{Z}/p\mathbb{Z})$ corresponding to the cover, denoted by $\mathrm{Mod}(S,[\beta])$. For $p=2$, the abelianizations of these groups were computed by Sato. We compute their abelianizations for every odd prime $p$ and show that they exhibit a splitting phenomenon different from the case $p=2$. In most cases, this difference is reflected in the image of the Prym representation; in the remaining cases, it is detected by the existence of a distinguished element in the Johnson kernel.
Let $X = \mathbb{P}^1 \smallsetminus \{0,1,\infty\}$ be the thrice-punctured over a ring of $S$-integers $\mathcal{O}_{K,S}$ in a number field~$K$. For any quotient $\pi_1^{\mathrm{mot}}(X,0) \twoheadrightarrow \Pi$ of Deligne--Goncharov's motivic fundamental group there is an associated Selmer scheme which parametrises $\Pi$-torsors with a mixed Tate motive structure. We give several descriptions of the motivic Selmer scheme which make it amenable to computations, using $\mathbb{G}_m$-equivariant cocycles of algebraic groups, Lie algebras, and complete Hopf algebras. We prove that the Selmer scheme is isomorphic to an affine space $\mathbb{A}^N_{\mathbb{Q}}$, and we construct coordinates realising this isomorphism. This is a key ingredient for making the motivic Chabauty--Kim method explicit in a general setting, without restrictions on the base field or the choice of fundamental group quotient.
Let $\mathsf k$ be a field, $S=\mathsf k[x,y,z]$, and $R=S/I$ be a standard graded Artinian Gorenstein $\mathsf k$-algebra of codimension three. The $h$-vector of such an algebra is known to be symmetric and unimodal. Mir\'o-Roig proved that if $\mathsf k$ is algebraically closed of characteristic zero and the $h$-vector of $R$ has at least three peaks, then $R$ has the weak Lefschetz property. In this article, we extend this result to any infinite field of arbitrary characteristic, using a different, elementary, and more direct argument. In particular, we recover Mir\'o-Roig's theorem without the hypothesis that $\mathsf k$ is algebraically closed. Along the way, we also prove a statement of independent interest that holds over any field: if the $h$-vector of $R$ has at least two peaks, and if $s$ is the largest degree of a peak, then the elements of $I$ of degree at most $s$ have no common factor.
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the classical Cauchy and dual Cauchy identities. We pose the problem of giving a direct combinatorial proof of these formulas for $0<k<\min\{m, n\}$. When $k=1$ or $k=\min\{m, n\}-1$, we give such a proof via an explicit sign-reversing involution on certain sets of Littlewood--Richardson tableaux.
Let $H_\Gamma$ be the Bestvina-Brady group associated to a finite connected graph $\Gamma$. For a biconnected defining graph, we prove two structure theorems. First, restriction induces an isomorphism $\mathrm{IAut}(A_\Gamma)\cong \mathrm{IAut}(H_\Gamma)$ compatible with the Andreadakis-Johnson filtrations. Second, the quadratic and cubic lower-central relation spaces, together with the separator arrangement detected by the Bieri-Neumann-Strebel invariant, determine a rational associative algebra $\mathscr{C}_\Gamma$. Every integral rank-one square-zero element of this algebra is realized by an automorphism of $H_\Gamma$, and the subgroup generated by these roots has finite index both in the cohomological image of $\mathrm{Aut}(H_\Gamma)$ and in the unit group of an integral order in $\mathscr{C}_\Gamma$. For an arbitrary connected graph, the graph-block decomposition gives the Grushko decomposition of $H_\Gamma$. Relative free-product automorphism theory then implies that $\mathrm{Aut}(H_\Gamma)$ and $\mathrm{Out}(H_\Gamma)$ are finitely generated and satisfy the Tits alternative relative to virtually polycyclic groups. We prove that $\mathrm{Aut}(H_\Gamma)$ is finitely presented if and only if $\mathrm{Out}(H_\Gamma)$ is finitely presented. This equivalence fails for higher finiteness properties without additional hypotheses: for $\Gamma_m=C_m\vee K_3$ with $m\geq 5$, $\mathrm{Out}(H_{\Gamma_m})$ is of type $F_\infty$, whereas $\mathrm{Aut}(H_{\Gamma_m})$ is of type $F_3$ but not $F_4$. We also construct a type-$F_\infty$ Bestvina-Brady group whose automorphism and outer automorphism groups are finitely generated but not finitely presented, and show that $H_{C_n}$ is not finitely presented for $n\geq 5$, whereas $\mathrm{Out}(H_{C_n})$ is virtually infinite cyclic.
Let $n\geq5$ and $m\geq3$. Let $\Phi\colon\mathrm{B}_n\to\mathrm{B}_m$ be a homomorphism of braid groups. We prove that if the image of $\Phi$ is irreducible and not cyclic, then $m=n$ and $\Phi$ agrees with an automorphism modulo the center $Z(\mathrm{B}_m)$. This resolves in the affirmative a conjecture of Chen, Kordek, and Margalit. It also provides a partial resolution to a problem on the K3 problem list. As a consequence, we prove that every holomorphic map $\mathrm{UConf}_n(\mathbb{C})\to\mathrm{UConf}_m(\mathbb{C})$ for $n\geq5$ and $m\geq3$ is affine equivalent to either a constant map or the identity map. This resolves a conjecture of Farb for $n\neq4$.
Rodrigo de Pool, Peter Huxford, Daniel Minahan et al.· 0 citations
