Papers

Let theta be an ordinary character or a Brauer character of a finite group N. We show that theta extends to every overgroup $G=G_\theta\trianglerighteq N$ if and only if Z(N) <= Ker(theta) and the deflated character of Inn(N) extends to Aut(N)_theta. This equivalence is compatible with fields of values. A similar investigation for extensions from subnormal subgroups leads to a new characteristic subgroup of N.

For $n\geq3$, let $\overline B_n=B_n/[P_n,P_n]$ be the crystallographic quotient of the Artin braid group, with translation subgroup $\overline P_n=P_n/[P_n,P_n]\cong\mathbb Z^N$, where $N=\binom n2$. We prove that every compact connected flat $3$-manifold $M$, considered up to affine equivalence, is the base of a geometric torus bundle whose total space is a compact flat manifold with fundamental group a Bieberbach subgroup of some $\overline B_n$. More precisely, for every $n\geq n_M$, where $n_M\in\{3,4,5\}$ is an explicit optimal threshold depending on $M$, we construct a Bieberbach subgroup $\Gamma_n(M)\leq\overline B_n$ of dimension $N$ and a complete normal subgroup $K_n(M)\cong\mathbb Z^{N-3}$ consisting of translations such that $$ \Gamma_n(M)/K_n(M)\cong\pi_1(M). $$ We also determine which of the associated torus-bundle group extensions split: for the realizations constructed here, splitting occurs for exactly four of the ten affine-equivalence classes of compact flat $3$-manifolds.

Let $G=C_2\times C_2$ and $A=\{0,1\}$. We characterize the submonoid of $\text{CA}(G;A)$ generated by lazy and invertible cellular automata. A cellular automaton belongs to this submonoid if and only if, whenever it maps the two free $G$-orbits onto the same free orbit, it maps both constant configurations and some two-point orbit to the same constant. The proof gives a factorization using elementary orbit collapses and three two-cell lazy maps. The criterion yields the cardinality $59\,712$ and identifies the maps missed by a weaker constant-separation invariant. Adjoining one idempotent generates the full monoid, and the generated submonoid is contained in exactly one maximal proper submonoid, of cardinality $65\,024$. The submonoid generated by lazy maps alone has $4\,007$ elements, and for both generating sets the maximum shortest-word length is six. The criterion and the cardinalities $59\,712$ and $4\,007$ are formally verified in Lean.

We study the finite basis problem for flat semirings of the form $S(W)$, where $W$ is a nonempty set of nonempty words, possibly infinite. We prove that $S(W)$ is finitely based if and only if the strict interior of every word in $W$ is linear; here the strict interior of a word is obtained by deleting its first and last letters. This completely resolves an open problem raised by Jackson et al.~(J. Algebra 611: 211–245, 2022).

For quandle rings, Bardakov–Passi–Singh and Elhamdadi–Nunez–Singh respectively formulated an analogue of Kaplansky's zero divisor conjecture and a weakened analogue of Kaplansky's idempotent conjecture involving the augmentation map. In this note, we verify these conjectures for medial quandles. The latter holds for a larger class of quandles; we characterize integral domains over which the statement holds for this class. We also characterize integral domains of characteristic 0 over which the statement holds for a given finite Alexander quandle.

Let $S=(\N,\max,+)$ be the max-plus algebra of the nonnegative integers, and let $\overline S$ be obtained from $S$ by adjoining $-\infty$ as a semiring zero. We first prove that the varieties generated by the full matrix semirings $\Mat_n(S)$ and by the upper triangular matrix semirings $\UT_n(\overline S)$ form strictly ascending chains. Every interval between distinct members of either chain has cardinality $2^{\aleph_0}$ and contains a chain and an antichain of that cardinality. The proof uses finite syntactic semirings associated with prime-cycle languages and an independent family of identities. We then construct finite commutative ai-semirings from languages of monomials closed under divisibility and prove their bounded-variable property by cancellation, explicit integer evaluations and independence in a free abelian group. A loop-block identity for upper triangular matrices and dimension-independent cubic-padding identities for full matrices show that every variety in each of the above intervals has infinite axiomatic rank. In fact, every variety between $\V(S)$ and the variety generated by all full matrix dimensions has infinite axiomatic rank. These varieties are therefore nonfinitely based, also relative to the ai-semiring axioms. Finally, for every fixed full matrix dimension, we prove equational equivalence over all six positive or nonnegative integer, rational and real coefficient domains. The original convex-ideal countermodels and their full-matrix transfer proof are recalled separately for comparison; they are not used in the algebraic proofs. Throughout, the signature consists of two binary operations and contains no constant symbols.

We establish new restrictions on the virtual cohomological dimension of word hyperbolic groups admitting Anosov representations into $\operatorname{PO}(p, q)$ ($p > q \geq 2$) or the exceptional Lie group $\operatorname{G}_2'$. Our main result states that, if $P_k$ is the parabolic subgroup of $\operatorname{PO}(p, q)$ stabilizing an isotropic $k$-plane in $\mathbb{R}^{p, q}$, a group admitting a $\{P_1,P_q\}$-Anosov representation into $\operatorname{PO}(p, q)$ has virtual cohomological dimension strictly less than $p$. As a result, a $\mathbf{H}^{p, q-1}$-convex cocompact subgroup of $\operatorname{PO}(p, q)$ cannot act properly discontinuously and cocompactly on $\mathbf{H}^{p, q-1}$. The main theorem also implies that a group admitting a $P_{\alpha}$-Anosov representation into the exceptional Lie group $\operatorname{G}_2'$ has virtual cohomological dimension strictly less than $4$, where $\alpha$ is a short restricted root of $\operatorname{G}_2'$. As an application, we prove that the homogeneous space $\operatorname{G}_2'/ \operatorname{SU}(2, 1)$ does not admit any compact quotient.

Let $G$ be a group of permutations of a set $\Omega$. The relational complexity $\mathrm{RC}(G)$ is the least integer $r\geq 2$ such that the $G$-orbit of every finite tuple is determined by the $G$-orbits of its $r$-subtuples; there is currently only a small collection of groups $G$ for which $\mathrm{RC}(G)$ is known precisely. In this paper we study the relational complexity of finite affine groups acting naturally on a $d$-dimensional vector space over $\mathbb{F}_q$. We prove that every such group has relational complexity at most $d+2$ when $q>2$ and at most $d+1$ when $q=2$; we produce a complete classification of the primitive such groups for which these bounds are attained.

We study maximal subgroups of topological full groups of irreducible one-sided shifts of finite type whose associated groupoids have Cantor unit space. For each wide open subgroupoid of such a groupoid, we construct a subgroupoid graph and identify the subgroupoid with a full compact open reduction of the corresponding countable graph groupoid. Every minimal wide open subgroupoid is either AF or purely infinite. A proper wide open subgroupoid is prime if it admits no proper intermediate wide open subgroupoid. We prove a trichotomy for prime subgroupoids: they are minimal, arise from a clopen bipartition, or have a unique nonempty proper closed invariant set, which is nowhere dense. For a prime subgroupoid, the normalizer of its topological full group is maximal in the ambient topological full group precisely when it maps onto the ambient abelianization. An analogous criterion holds in every subgroup containing the ambient commutator subgroup. For a minimal prime subgroupoid, its topological full group is maximal precisely when it equals its normalizer and the inclusion induces a surjection on first groupoid homology. We also describe all fully amplified maximal subgroups, namely those closed under local agreement. For Thompson's group $V$, the stabilizer of every simple type system is maximal, extending results of Belk, Bleak, Quick and Skipper for finite type systems and certain classes of infinite ones. Finally, every subgroup containing the ambient commutator subgroup has continuum many pairwise nonisomorphic maximal subgroups acting minimally on the unit space.

We obtain a uniform lower bound for the drift of random walks on hyperbolic spaces in terms of joint minimal displacement. Our main theorem applies to weakly acylindrical actions on Gromov-hyperbolic spaces and is quantitative in all relevant constants. For discrete isometry groups of Hadamard manifolds with pinched negative curvature, the required assumptions are automatically satisfied by the Margulis lemma. This yields uniform gaps at zero in drift spectra for pinched negatively curved Hadamard manifolds, hyperbolic groups, and mapping class groups acting on curve graphs.

Let $(W,S)$ be a Coxeter system of finite rank, let $T$ be its set of reflections and let $c \in W$ be a Coxeter element. For $x$ in the interval $[1,c]_T$ of the absolute order, let $P(x)$ be the subgroup generated by the reflections $t\le_T x$. We prove that $x$ is a Coxeter element of $P(x)$ with respect to the canonical Coxeter generators of $P(x)$, that $P(x)$ is generated by the factors of any reduced reflection factorization of $x$, and that the Hurwitz action on the reduced reflection factorizations of $x$ is transitive. The proof reduces the general case to the case of a Coxeter system whose Coxeter matrix has entries in $\{1,2,3,\infty\}$, which was treated by Hubery and Krause.

We provide a sufficient condition for a group of homeomorphisms of a compact Hausdorff space to be non-invariably generated, generalizing previous results proved using the dynamics of a group's action. We exhibit new applications among Scott-Röver-Nekrashevych groups and among groups of tree automorphism of with (almost) locally prescribed actions.

We study the Ahlfors-regular conformal dimension of spaces with locally expanding self-covering maps and, more generally, limit spaces of contracting self-similar groups. We show that the conformal dimension is characterized by combinatorial moduli associated with the Schreier graphs of these groups. We compare these moduli with contraction coefficients of self-similar group actions and with algebras of functions on the limit spaces satisfying asymptotic $p$-energy conditions. We also give conditions on the group when the conformal dimension is equal to the infimum of visual metrics on the self-similarity complex.

Let $G$ be a finite group, $M$ a $G$-module, and $F$ a field. A continuous group homomorphism $i : \Gamma_L=\text{Gal}(L_{\text{sep}}/L) \rightarrow G$ yields a homomorphism $i^* : H^d(G,M) \rightarrow H^d(L,M)$ for every $d\in \mathbb{N}$. An element $u \in H^d(G,M)$ is negligible over $F$ if $u\in\ker(i^*)$ for every $i$ and every field extension $L/F$. If $M=\mathbb{F}_p$ for a prime $p$ with a trivial $G$-action, an element $u$ is eventually negligible over $F$ if $u^k$ is negligible over $F$ for some $k\geq1$. The (eventually) negligible elements over $F$ form an ideal of the cohomology ring $H^*(G,\mathbb{F}_p)$, called the (eventually) negligible ideal. We prove an analog of Quillen-Venkov for eventually negligible elements. An element $u\in H^*(G,\mathbb{F}_p)$ is eventually negligible over $F$ if and only if the restriction of $u$ to the cohomology of each elementary abelian $p$-subgroup of $G$ is eventually negligible over $F$. Paired with applications of a Gysin-type long exact sequence, we identify generators of the negligible ideal over $\mathbb{Q}$ of the mod-2 cohomology of dihedral groups, cyclic $2$-groups, and generalized quaternion groups. We prove that the negligible ideal over $\mathbb{Q}$ of the mod-2 cohomology of every finite irreducible Coxeter group is radical. The result was previously known for Weyl groups; we prove it for the remaining non-Weyl types $I_2(m)$, $H_3$, and $H_4$.

Let $f,g\in\mathbb{Z}[t]$ be products of cyclotomic polynomials such that either $g/\gcd(f,g)$ or $f/\gcd(f,g)$ (or both) is squarefree. We give explicit formulae for the invariant factors of $f(C_g)$, where $C_g$ is the companion matrix of $g$, expressing them in terms of the indices and multiplicities of the cyclotomic factors of $f$ and $g$. A more compact formula holds for the circulant case $g(t)=t^n-1$, assuming that every cyclotomic factor shared by $f$ and $g$ has multiplicity $1$ in $f$. We discuss applications to several topics, including abelianizations of cyclically presented groups, homology groups of cyclic branched covers of knots, adjacency matrices and shifted graph Laplacians, $K$-theory, and Smith forms of matrix powers. For example, for two families of hyperbolic knots we give the first integral homology in every covering degree; or, when $g$ is squarefree, we determine the least eventual period of the Smith forms of powers of $f(C_g)$.

We compute the Euler characteristic of Bestvina-Brady kernels $\operatorname{BB}(\Gamma)$ for all Artin groups of spherical type. Our method involves discrete Morse theory to describe the cellular structure of a classifying space for the Bestvina–Brady kernel, through the action of $\operatorname{BB}(\Gamma)$ on non-integer level sets of the universal cover of the Salvetti complex. We extend the method of computation of $\chi(\operatorname{BB}(\Gamma))$ to Artin groups that satisfy the $K(\pi,1)$-conjecture and whose nerves are rationally acyclic, and we provide an example by realising the projective plane as the nerve of a Coxeter group.

We prove that if M is an abelian group and X = Alex(M, phi) is a Latin Alexander quandle with id - phi in Aut(M), then Aut(X) is isomorphic to M semidirect C_Aut(M)(phi), Inn(X) is isomorphic to M semidirect <phi> and Out(X) is isomorphic to C_Aut(M)(phi)/<phi>. As a consequence, we obtain a description of the automorphism groups of direct products of the cyclic commutative medial quandles C_{2n+1} and the dihedral quandles R_{2n+1}, and we find their orders. Furthermore, we correct a result of Bardakov-Fedoseev [Algebra and Logic 63 (2024), no. 2, 111-142] on multiplication of quandles and prove that the Borodin-Neshchadim-Simonov and Markhinina-Nasybullov quandles can be constructed from conjugation, core, and generalized Alexander quandles, using the operation of multiplication of quandles. Finally, we consider the quotients of a quandle Q, namely the medialization Q^med, the commutativization Q^com, and the trivialization Q^tr. We prove that the conjugation quandle Conj(G) is medial if and only if gamma_2(G) <= Z(G); we find bounds for |Conj(G)^med| and prove that Core(G)^com is isomorphic to Core(G/<<g^3>>). We describe commutative generalized Alexander quandles and, as a consequence, find the commutativization of the dihedral quandle: R_n^com is isomorphic to R_gcd(n,3).

Let $G$ be a group and let $A$ be a set with at least two elements. A lazy cellular automaton over $A^G$ acts as the identity except at the cells where a fixed pattern occurs, where it replaces the central symbol by a fixed different symbol. Let $M(G;A)$ be the submonoid of $\mathrm{CA}(G;A)$ generated by the lazy and the invertible cellular automata. We study when $M(G;A) = \mathrm{CA}(G;A)$. Since no lazy cellular automaton is surjective, every surjective element of $M(G;A)$ is invertible; hence $M(G;A) \neq \mathrm{CA}(G;A)$ whenever $A$ is infinite or $G$ has an element of infinite order. Moreover, if $|A| = 2$ and $G$ has a quotient isomorphic to the Klein four-group $C_2 \times C_2$, then $M(G;A) \neq \mathrm{CA}(G;A)$. For locally finite $G$, and in particular for finite $G$, these are the only obstructions: we prove that $M(G;A) = \mathrm{CA}(G;A)$ if and only if $3 \leq |A| < \infty$, or $|A| = 2$ and $G$ has no quotient isomorphic to $C_2 \times C_2$.

We prove that every Lie group can be realized by permutations of a countable set, answering a classical question of Ulam. The key deformation construction originated in an argument produced by OpenAI's GPT 6 Astra during extended mathematical discussions.

We introduce the leap group of a graph, the subgroup of the symmetric group on the vertex set generated by graph derangements. We summarize historical results on graph derangements, which we call hops, and prove additional existence and complexity results. The leap groups of various canonical graphs are computed, and their behavior under basic graph operations is analyzed. We offer partial results on which groups can appear as leap groups, and study hops and leaps for infinite and directed graphs.

The goal of this note is to provide a human written exposition of OpenAI's recent proof that Thompson's group F is nonamenable.

We extend Burnside's $p^a$-Theorem on conjugacy classes of prime-power size by proving that no non-abelian finite simple group possesses a conjugacy class of size $2p^n$ for any prime $p$ and $n\geq 0$. Based on this, we partially extend a well-known result of L. Kazarin by showing that, given a finite group $G$, the subgroup generated by a conjugacy class $x^G$ of size $2p^n$ where $x$ has odd order, is a solvable normal subgroup of $G$. If instead the subgroup $\langle x^G \rangle$ generated by such a class $x^G$ is non-solvable (which forces $x$ to have even order), then $p= 5$ and $\langle x^G \rangle/S(\langle x^G \rangle)$ is isomorphic to the symmetric group $S_5$, where $S(\langle x^G \rangle)$ denotes the solvable radical of $\langle x^G \rangle$.

We study learning an unknown element $g$ of a group acting with finite orbits on a countable set, from examples $(x,gx)$. After a sample on $A$, the consistent hypotheses are the restrictions of a coset of the pointwise stabilizer of $A$, and the sample forces exactly the labels of points this stabilizer fixes; for a Galois group, they form the field generated by the data. Through this correspondence, multiclass dimensions become group invariants: the DS and graph dimensions equal the height, the Littlestone dimension the longest irredundant sequence length, and the teaching dimension the minimal base size. Among deterministic learners, symmetry makes empirical risk minimization optimal: on nondegenerate classes every consistent learner, whatever its tie-breaking, attains the optimal realizable sample complexity up to constant factors, and every consistent online learner the optimal mistake bound. Agnostically, every ERM's sample complexity exceeds the known lower bound by at most the DS-to-Natarajan ratio, up to constants. For finite Galois extensions of countable fields, the Natarajan dimension equals a group invariant, the coset-cube number, which is at least the number of complemented factors of any chief series of any subgroup. Hence the dimensions coincide for every solvable Galois group, while for the groups $A_5$ and $S_5$ of the quintic they are $2<3$ and $3<4$, and $A_5$ is the smallest group with a gap. For the Galois group $S_n$ of the general equation of degree $n$, they coincide exactly when $n\le 4$, i.e., in characteristic zero, exactly when the equation is solvable by radicals; the implication is one-way, as the simple group $\mathrm{PSL}(2,8)$ has no gap. For linear, orthogonal and unitary maps of $\mathbb R^d$ or $\mathbb C^d$ both dimensions equal $d$, and a two-fibered pseudo-cube is a disjoint union of Natarajan cubes exactly when its monodromy group is abelian.

We prove that for every $\varepsilon >0$, there is $\delta >0$ such that for every probability measure $\mu$ on a free group $\mathbb{F}$, if the operator norm of the action of $\mu$ on $L^2\mathbb{F}$ by convolution exceeds $\varepsilon$, then there is a coset of an abelian subgroup of $\mathbb{F}$ with $\mu$-mass at least $\delta$.

We propose the study of the class of groups that fix a point for every continuous action on any convex compact space. Oliver completed the corresponding program of classifying the finite groups that have this fixed-point property in finite dimensions. Turning to infinite groups, the Schauder–Tychonoff theorem establishes that the integers have the fixed-point property, but this has not been shown for any other infinite, finitely generated group. We establish a local transfer principle and use it to show, as a proof of concept, that $\mathbf{Z} \times (\mathbf{Z}/p\mathbf{Z})$ belongs to this class of groups.

Let $\X$ be a variety of groups. A group $G$ is $\CSX$ if its maximal $\X$-subgroups are malnormal. We prove that the amalgamated free product of two $\CSX$-groups with a malnormal amalgam and without any involutions is again $\CSX$. Using this, we describe conditions under which the HNN-extension of a $\CSX$-group is $\CSX$.

We give two types of examples where two non isomorphic finite quandles have isomorphic enveloping groups. This implies that two non-isomorphic finite quandles can share isomorphic representation categories (as additive and tensor categories simultaneously).

Let $n,k$ be positive integers and $q$ be a prime power. We prove that there exists a real constant $C_k$ depending only on $k$ such that if $ \# M_n(\mathbb{F}_q) > C_k$ then every element in $M_n(\mathbb{F}_q)$ is a sum of two $k$th powers, proving Larsen's conjecture.

A recent preprint constructs a finitely presented torsion-free group $G$ with a finite two-dimensional classifying space such that $\mathbb F_2[G]$ contains nonzero zero divisors. We extend this to every odd prime $p$: there is a finitely presented torsion-free group $G_p$, again with a finite two-dimensional classifying space, such that $\mathbb F_p[G_p]$ contains nonzero elements $\alpha,\beta$ with $\alpha\beta=0$. Together with the case $p=2$, Kaplansky's zero-divisor conjecture therefore fails over every field of positive characteristic. The new ingredients are a signed version of the parity cancellation, made possible by a bipartite random model on one side; a type design built from the projective plane over $\mathbb F_{p^3}$ with $p$ additional letters, which makes every cross-intersection congruent to $1$ modulo $p$; and a squared-word decay estimate for this design, proved with an explicit test vector. The geometric part of the argument is taken from the characteristic-two construction. The characteristic-zero case remains open.

A complete mapping of a finite group $G$ is a permutation $\theta$ of $G$ such that $x\mapsto x\theta(x)$ is also a permutation of $G$. Complete mappings have applications in cryptography, particularly in the design of round functions for cryptographic systems. The study of their parity has led to the question of which finite groups admit complete mappings of both parities. A useful approach to this question is to construct complete mappings stabilizing suitable subgroups, in particular Sylow $2$-subgroups. We show that, for every $n\geq2$, every prime power $q$, and every $Z\leq Z(\mathrm{SL}_n(q))$, the group $\mathrm{SL}_n(q)/Z$ admits a complete mapping stabilizing a Sylow $2$-subgroup if and only if $(n,q)\neq(2,2)$.