Papers

Motivated by the work of Druţu–Sapir, we introduce a definition of relatively hyperbolic TDLC groups that depends only on their Cayley-Abels graphs and show that this new definition is equivalent to one introduced by Arora–Martínez-Pedroja. Then, we prove that relative hyperbolicity for TDLC groups is invariant under quasiisometry.

Let $F$ be the free group of rank two and, for a prime $p$, let $\mathbf H_p=\mathbf G_p*\mathbf{Ab}_{p-1}$ be the pseudovariety of finite groups having a normal $p$-subgroup with abelian quotient of exponent dividing $p-1$. For $H$ of rank two with $\ab_F(H)=F^{\ab}$ we determine the set $\mathfrak D(H)$ of primes $p$ where $H$ is $\mathbf H_p$-dense in $F$, and we show that this prime set is Frobenian in the sense of Serre: it is governed by a single Laurent polynomial $g$ attached to $H$. We prove that $p\in\mathfrak D(H)$ if and only if $g$ has no zero on the torus $(\F_p^\times)^2$, and hence that $\mathfrak D(H)$ possesses a computable natural density $d(H)$. Exactly one of three cases holds: $\mathfrak D(H)$ is the set of all primes, it is finite, or it is neither and $d(H)$ satisfies $\frac1{|G|}\le d(H)\le1-\frac1{|G|}$, where $G$ is the finite Galois group attached to $g$.

We obtain explicit formulas for heat kernels of the normalized Laplacians for two infinite families of Cayley graphs. The first family corresponds to free products of finite groups $G=G_1*\dots*G_r$, with $r\ge2$, and where all factors $G_i$ have the same order $L\ge2$. The second family corresponds to free products of two arbitrary nontrivial finite groups $G*H$ of unequal order. In both families, the generating set consists of all non-identity elements in each factor. Extending the approach of Chung–Yau, we show that the Cayley graph in each family strongly and regularly covers a weighted half-line and a weighted line, respectively. We solve the spectral problems on the half-line and the line, and then apply spectral transfer principles for strong and regular coverings to derive formulas for the heat kernels on the Cayley graphs. For the first family, the spectrum of the Laplacian consists of a single interval together with, when $L>r$, one additional isolated eigenvalue. When $L=2$, we recover the well-known results for regular trees. For the second family, the spectrum of the Laplacian consists of two intervals and two eigenvalues.

Graphical discreteness is not preserved under quasi-isometry in general. We prove that, for every finite thick generalized $m$-gon $\Gamma$ with $m\geq 3$, graphical discreteness is nevertheless constant on the class of finitely generated groups quasi-isometric to the right-angled Coxeter group $W_\Gamma$: such a group $\Lambda$ is graphically discrete if and only if $\Gamma$ is nonflexible, meaning that no nontrivial graph automorphism of $\Gamma$ fixes a closed star pointwise. The proof combines quasi-isometric rigidity of Fuchsian buildings with a metric-strata argument that recovers the standard Coxeter Cayley graph from the building metric. For finite-rank Coxeter systems, we also show that the automorphism group of the standard Cayley graph is compact-by-discrete if and only if it is discrete, equivalently, if and only if the defining diagram is nonflexible. We further obtain obstructions to graphical discreteness from one-vertex doublings, classify compact-by-discreteness for standard chamber graphs of graph products of finite groups, and derive applications to finite projective planes and to right-angled Coxeter groups with Menger curve boundary. Our methods do not settle whether nonflexible finite thick generalized polygons exist.

Let $G$ be a finite solvable group such that, for any prime $p$ and any quotient $Q$ of $G$, there are two Sylow $p$-subgroups of $Q$ intersecting in $O_p(Q)$. Then, for every family $(P_i)_{i=1}^n$ of Sylow subgroups of $G$ for distinct primes $p_1,\ldots,p_n$, there exists $x \in G$ such that $P_i \cap P_i^x = O_{p_i}(G)$ for all $i$. This covers groups of odd order, partially settling a conjecture of the second and fourth authors and unifying old results of Bialostocki and Mann on the intersection of nilpotent subgroups. We also prove a result for all symmetric and alternating groups, completing the proof of the conjecture for simple groups as initiated by Burness and the first author.

For a subset $S$ of a finite group $G$, let $I_G(S)$ denote the set of commutators $[g,x]=g^{-1}g^x$, where $g\in G$ and $x\in S$. Suppose that a finite group $G$ has a Carter subgroup $C$, that is, a nilpotent subgroup containing its normalizer. Suppose that any subgroup generated by a subset of $I_G(C)$ is $r$-generated. We prove that if $G$ is soluble, then the derived subgroup $G'$ has $r$-bounded rank. We produce examples showing that the solubility condition cannot be dropped. For any finite group, we prove that the rank of $G'$ is $(r,l)$-bounded, where $l$ is the maximum rank of composition factors of $G$ isomorphic to $PSL_2(q)$ for $q\equiv 7\,(\operatorname{mod}8)$. We also prove in the general case that $G'$ has $r$-bounded rank under the additional condition that for any $x\in I_G(C)$, any subgroup generated by a subset of $I_G(x)$ is $r$-generated. The proofs rely on the classification of finite simple groups, using which we prove that if a finite simple group $G$ has an element $x$ of prime order $p$ such that any subgroup generated by a subset of $I_G(x)$ is $r$-generated, then $G$ has $r$-bounded (Prüfer) rank or is isomorphic to $PSL_2(q)$ with $q \equiv 3\,(\operatorname{mod}4)$ when $p=2$, or to $PSL_2(q)$ with $q \equiv -1\,(\operatorname{mod} p)$ when $p\ne 2$.

For each $g\ge 2$, we show that there are finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds with genus $g$ fiber. More generally, we show that there are only finitely many conjugacy classes of admissible surface subgroups of PSL$_2(\mathbb{C})$ whose image is contained in the fundamental group of some arithmetic hyperbolic $3$-manifold. We also give an effectively computable upper bound with effective asymptotic rate on both finiteness statements. This affirms a conjecture of Bowditch, Maclachlan, and Reid.

A connected skew brace of finite Morley rank is left nilpotent (resp. weakly soluble) whenever the additive and multiplicative groups are nilpotent (resp. are soluble and the left chief length is at most 3).

We prove that $S^1$, ${\rm SU}(2)$, and ${\rm SO}(3)$ are compact amalgamation bases: for each of these groups, any two compact groups with specified embedded copies of it embed in a compact group in which those copies agree. For compact Lie groups, the amalgam can be chosen to be a finite-dimensional unitary group. This research was conducted with substantial AI assistance, including in the development of mathematical constructions and proof arguments.

Using the idea of Venkataramana's construction of infinite families of arithmetic orthogonal hypergeometric groups, we construct some of the very first examples of infinite families of arithmetic symplectic hypergeometric groups that do NOT satisfy the arithmeticity criterion of Singh and Venkataramana. For example, we show that the hypergeometric groups associated to the pairs of polynomials $(x-1)^4P_m(x^{9})$ and $(x^4+x^3+2x^2+x+1)Q_m(x^{9})$, where $P_m$ and $Q_m$ are integral polynomials of degree $2m$ such that the corresponding pair determines a Zariski dense symplectic hypergeometric group, are arithmetic in $\operatorname{Sp}(18m+4)$ for any integer $m\in\mathbb{N}$.

We establish a sufficient condition for an additively idempotent semiring to be inherently nonfinitely based: if its generated variety is locally finite and every Zimin word is minimal in the additive order, then it is contained in no finitely based locally finite variety. We characterize Zimin minimality by the membership of a countable flat factor semiring $\Finf$ in the generated variety and prove that $\V(\Finf)$ is a minimal inherently nonfinitely based variety. We also obtain general restrictions on the additive order of finite inherently nonfinitely based ai-semirings. As applications, the six-element ai-semirings $\A$ and $\B$ are shown to be inherently nonfinitely based. Finally, we prove that the six-element ai semiring $\Abar$ is not inherently nonfinitely based and is finitely based. Its multiplicative reduct is inherently nonfinitely based; thus, to the best of our knowledge, $\Abar$ is the first finitely based ai-semiring with an inherently nonfinitely based multiplicative reduct.

We give a unified treatment of definite Albert algebras over $\mathbb{Z}$, with exceptional Siegel–Weil identities as the main arithmetic tool. Starting from the standard octonion order alone, its first scalar theta coefficient saturates the genus average and forces every remaining class to be rootless. A cubic-ring coefficient of the $G_2$ theta lift then gives the mass of the remaining scale-$2$ framed orders, also recovered from shell geometry. A uniform Niemeier–Golay stabilizer bound saturates this mass, proving that there is exactly one further class and determining both automorphism orders, frame transitivity, and the $2457$ frames on $819$ rank-one elements. The associated generalized hexagon identifies the nonstandard group as ${}^3D_4(2):3$ only after its order has been proved. An independent weighted-theta argument gives the trace-zero root-system alternative $\varnothing$ or $A_2E_8^3$. The construction relates positive masses of framed objects to exceptional Eisenstein coefficients.

In this note we develop a novel approach for proving the non-vanishing of bounded cohomology. This utilizes a splitting argument whose simplest form is as follows: Let M denote an n-manifold of non-zero simplicial volume and N a codimension two submanifold of M, then one can conclude that the n-th bounded cohomology of the fundamental group of M \ N is non-zero. We then translate the existence of a complement with a given fundamental group into an easily accessible homology computation, which might be of independent interest.

We develop a geometric lifting method for inverse sumset problems in compact connected abelian groups. The first result is an analogue of Freiman's $3k-4$ theorem, that is every compact set $A\subseteq G$ of sufficiently small Haar measure satisfying $\mu_G(A+A)<3\mu_G(A)$ is contained in a one dimensional Bohr set of measure at most $\mu_G(A+A)-\mu_G(A)$. This resolves a question of Christ and Iliopoulou. The proof combines Bilu's theorem with a geometric refinement of the spillover argument. We also establish a sharp projection theorem. Under a continuous surjective homomorphism with connected kernel, a compact set of sufficiently small positive measure and doubling at most $K$, where $2\le K<3$, has image of doubling at most $2K-2$, and this factor is best possible. Further consequences include variants of the $3k-4$ theorem for popular sumsets and an inverse theorem for Tao's convolution inequality.

Let H be a linear group. We show that if there is an invertible matrix x such that all the elements of xH share the same characteristic polynomial then H is virtually solvable. There are plenty of applications that will be presented in future paper. Here, we discuss some applications to the generalized Weigold conjecture and present an alternative straightforward proof of the Formanek–Procesi nonlinearity theorem for Aut(F_n), n>2, over every field. When n>4 our non-linearity proof gives a stronger result than the original Formanek–Procesi theorem.

We show the smallest non-abelian quotients of spherical and affine Artin groups are isomorphic to the smallest non-abelian quotients of the corresponding Coxeter groups. We deduce irreducible spherical Artin groups are determined by their finite quotient groups.

We prove that the three-element additively idempotent semiring $S_7$ is strongly nonfinitely based: every finite additively idempotent semiring whose generated variety contains $S_7$ has no finite equational basis. This answers Problem~1.2 in Yue, Ren and Gao's paper Explicit equational bases for the power semirings of $S_7$ affirmatively. The proof combines a rigidity property of set systems with private points and a probabilistic construction derived from semilinear sets of letter counts in finite semigroups. The argument allows arbitrary polynomial substitutions and does not assume commutative multiplication. As consequences, both power semirings of the multiplicative reduct of $S_7$, and both power semirings of every finite group of order at least three, are strongly nonfinitely based. All algebras are considered in the signature $(+,\cdot)$ without nullary operations.

We show that the six-element additively idempotent semiring $B_2^1$ is inherently nonfinitely based. More precisely, for every positive integer $n$, we construct an infinite additively idempotent semiring, generated by at most $(6n+8)^2$ elements, that satisfies every $n$-variable identity of $B_2^1$.

A conjecture of Lewis asserts that two reflection factorizations of the same element in a finite Coxeter group $W$ belong to the same Hurwitz orbit if and only if they generate the same subgroup $H$ of $W$ and have the same multiset of $H$-conjugacy classes. We prove this conjecture for every finite Coxeter group all of whose irreducible components are of type $A$, $B$ or $D$. Furthermore we prove a reduction, which reduces the conjecture to a finite problem for the exceptional types.

We present a computational framework to verify a refinement of the McKay conjecture, proposed by E. Giannelli, for sporadic simple groups and their universal covers. This refinement asserts the existence of a bijection between the $p'$-characters of a finite group and those of the normalizer of its Sylow $p$-subgroup that does not increase character degrees. Our verification enforces stricter conditions, ensuring equivariance under the relevant outer automorphism group, preservation of the $p$-part of the character conductor, and compatibility with central characters. We establish theoretical reductions that limit the relevant outer automorphism action to an order of at most 2, allowing us to leverage Clifford theory and precomputed character tables to circumvent expensive explicit group constructions. Finally, we provide algorithms for classifying characters and constructing bijections that strictly satisfy these conditions.

We prove Miller's conjectured bound for solvable groups: if $G$ is a finite solvable group and $\chi\in\Irr(G)$, then $\chi(g)$ is zero or a root of unity for at least half of the elements $g\in G$. We also prove the same bound for monomial irreducible characters of arbitrary finite groups.

A $\textit{magic-lined hypercube}$ (or, simply, $\textit{magic hypercube}$) of order $k$ and dimension $n$ is an arrangement of the numbers $1,\dots,k^n$ in a $k\times\cdots\times k$ ($n$-fold) grid such that every line of $k$ numbers parallel to a coordinate axis has the same magic sum. While such hypercubes exist for every order $k\ge 3$ and every dimension $n$, no magic-lined hypercube of order $2$ exists in any dimension $n\ge 2$. For hypercubes of order $2$, we thus relax the magic condition from lines to two-dimensional faces or planes. We call an arrangement of the numbers $1,\dots,2^n$ in a $2\times\cdots\times2$ ($n$-fold) grid magic-faced if every $2\times2$ face has the same magic sum. We prove that a magic-faced hypercube of order $2$ exists in every dimension $n\ge0$, and that it is unique up to a certain natural set of transformations of size $2^n(n+1)!$ when $n$ is even and $2^n n\cdot n!$ when $n$ is odd. As an application, we recover and generalize the classical $4\times4$ Khajuraho magic square, answer a question of Coxeter on the group acting on the $384$ “most-perfect" $4\times4$ magic squares, and extend the picture to higher dimensions. In particular, we prove that, in dimension $n$, these most-perfect objects form a single orbit under a certain natural action of the Weyl group $W(B_{2n})$.

We exhibit explicit ICC subgroups of $\SL_7(\mathbb Q)$ with Kazhdan's property $(T)$ that is not properly proximal.

We study the topological and geometrical aspects of non-determinism in group actions, generalizing the notion of non-expansive direction for abelian actions. Under suitable conditions on the acting group, we establish restrictions on the collection of non-deterministic horoballs (or half-spaces) for any action. We illustrate our results in the case of free and nilpotent groups, expanding upon known cases for abelian groups. We then apply these findings to standing problems concerning minimal self-joinings in topological dynamics, which are of independent interest.

As an extension of the It^o-Michler theorem to the nonabelian or nonnormal Sylow subgroups, A. Moreto raised the following conjecture: let G be a finite group and P a Sylow p-subgroup of G. Then the number of irreducible characters of G whose degrees are divisible by p is not less than the length of any subgroup chain between N_G(P) and G. In this short note, we give a counterexample to this conjecture.

Many numerical invariants of a finite group arise as the multiplicity of the trivial character in a naturally associated class function, and the remaining multiplicities carry finer structural information. For the number of conjugacy classes of elements whose order involves only a chosen set of primes, the natural such class function is a generalized character interpolating between the classical conjugating character and the regular character. We prove that it is a character in two important general cases, obtained independently by Robinson, and we rule out a family of groups and characters in which, we argue, a counterexample would be most likely to arise. Furthermore, we characterize when these class functions are compatible with passing to a subgroup and thereby sharpen a theorem of Sangroniz. The analogous class function for real elements is shown to always give a character. Finally, we give a dual construction that, together with the recently proved McKay Conjecture, yields a best-possible criterion for a finite group to have an abelian Sylow $p$-subgroup.

For positive integers \(m,n\), let \[ A_{m,n}=(\{1,\ldots,m\}\times\Z_n)\cup\{0\} \] have flat addition and multiplication truncated at degree \(m\). We prove that \(A_{m,n}\) is finitely based exactly when \(m\le 2\) or \((m,n)=(3,1)\). Explicit finite bases are supplied throughout this region. Outside it, high-girth hypergraphs with a constant-sum rigidity property yield finite countermodels to every bounded-variable fragment of the equational theory. The proof places no divisibility or coprimality restriction on the parameters.

We study additively idempotent semirings obtained from commutative words by equipping their subwords with the divisibility order. Every finite semiring associated with a power of one letter is finitely based, whereas one associated with a linear word is finitely based exactly when the word has length at most two. A hypergraph preservation lemma yields the nonfinite basis result and extends it to intervals of varieties. We establish a sharp containment criterion between the power and linear families, and determine the finite basis property of every join of two varieties generated by one member of each family. The unrestricted power family generates the nonfinitely based max-plus variety. In contrast, the unrestricted linear family has a finite basis, as does its join with each finite power member. We also realize a previously known six-element limit semiring as a quotient of a subsemiring of the eight-element linear-word semiring. This gives a proper nonfinitely based subvariety and resolves the corresponding minimality question.

The input for the binary diameter problem consists of explicitly represented permutations generating a finite group $G$ and a binary-encoded nonnegative integer $k$. The question is whether every element of $G$ is a product of at most $k$ input generators. For the binary length problem, the input contains in addition a permutation $g \in G$ and it is asked whether $g$ is a product of at most $k$ input generators. We prove that the binary diameter problem is PSPACE-complete. When restricted to $2$-step nilpotent groups, the binary diameter problem is shown to be complete for $\mathsf{\Pi_2^P}$, whereas the binary length problem is shown to be NP-complete. Without the restriction to $2$-step nilpotent groups, the binary length problem is PSPACE-complete by a result of Jerrum.

We give an explicit geometric and algebraic description of the index map for AF-by-discrete groupoids. The index records transport between infinite tiles and abelianized returns in the groups of germs, subject to a defect in the dimension group of the associated Bratteli diagram. We determine the kernel of the index map and construct transposition factorizations of zero-index elements with trivial germs at singleton orbits. In particular, in every minimal case, this kernel equals the subgroup generated by dynamical transpositions. We also characterize approximation by the fixed AF core and prove almost finiteness for minimal systems that are compactly generated. We make higher homology explicit in terms of the groups of germs. We express the secondary parity differential by lifted surface relations and by an extension class constructed from balanced boundary data. Under minimality and comparison, lifting finite relations gives a split AH sequence. Explicit boundary and matrix calculations, together with a nonminimal example with nonzero parity differential, illustrate the theory.