Papers

We study intervals in the lattice of additively idempotent semiring varieties associated with power semirings of finite groups. We prove that every variety lying above the nonempty power semiring of a finite group of order at least three and below a variety generated by finitely many full power semirings of finite groups is nonfinitely based. In fact, none of these varieties admits an identity basis with a fixed finite bound on the number of variables. In particular, adjoining the empty set gives an interval consisting entirely of nonfinitely based varieties. We also construct equational upper bounds that are closed under zero adjunction and yield further intervals with the same property. The proof combines finite commutative quotient semirings, a cardinality estimate for kernel blockers over finite modules, and a uniform estimate for fibres of ordered products in finite groups. As a consequence, the full power semiring of a finite group is finitely based precisely when the group has at most two elements.

All the eigenvalues of an integral graphs are integers. Integral graphs are extremely rare. They form an asymptotically vanishing fraction $2^{-\Omega(n)}$ among all graphs on $n$ vertices. It makes the construction of a new family of integral graphs a challenging task. Also, most of the known infinite family of integral graphs rely on Cayley graphs over Abelian groups. In this article, we introduce a new family of integral graphs obtained from the groups. The construction of our graphs from groups is different from the construction of Cayley graphs. A spectral uniqueness theorem is established, which shows that each member of the infinite family is determined by its adjacency spectrum among all finite simple graphs. We also present recursive constructions that generates larger members of the family from smaller ones, providing a scalable class of integral graphs. Finally, we investigate algebraic realizations of these graphs as complements of Proper Prime Order Element Graphs of finite $2$-groups and obtain conditions characterizing such realizations. We also observe that the graphs obtained from different non-isomorphic groups have cospectral graphs.

We prove that irreducible spherical type Artin groups are profinitely rigid within the class of all spherical type Artin groups. As part of the proof, building on ideas of Kolay for the $n$-strand braid group, we compute the smallest non-abelian quotient of every spherical type Artin group. In particular, given an irreducible spherical type Artin group, its smallest non-abelian quotient is always that of the corresponding Coxeter group, except for dihedral Artin groups of weight divisible by $4$, which all have $\mathfrak S_3$ as a non-abelian quotient. Parts of the proof also rely on the computation of the cohomological dimension of the profinite completion of certain Artin groups.

We prove that the variety generated by the max-plus algebra on the nonnegative integers has a unique limit subvariety, namely the variety generated by the six-element semiring constructed by Shao, Ren and Gao. More generally, we characterize hereditary finite basedness in a finitely defined variety of commutative additively idempotent semirings containing the max-plus variety. Its hereditarily finitely based subvarieties form a finite lattice and admit equational bases involving at most ten variables. We also prove that every proper subvariety of the max-plus variety is locally finite. A finite basis theorem for finite semirings with a multiplicative identity establishes finite basedness of all finite truncations. Finally, we determine the finite basis properties of two families of subvarieties defined by power identities and of subvarieties relatively defined by arbitrary families of identities making individual additive subterms greatest elements.

We solve the finite basis problem for all subvarieties of $\mathsf{V}(S^0_7)$. By considering their intersections with $\mathsf{V}(S_7)$, we reduce the problem to six intervals. We show that $\mathsf{V}(S_7^0)$ has exactly fifteen finitely based subvarieties and that every other subvariety is nonfinitely based. Moreover, $\mathsf{V}(S_c(abc))$ and $\mathsf{V}(SR_6)$ are the only limit subvarieties of $\mathsf{V}(S_7^0)$.

We study the (Haar) measure of verbal sets in profinite groups, and especially in free profinite groups. Relating the measure of a verbal set with the width of the corresponding word, we find verbal sets of positive measure different from one. We prove a sufficient condition for the measure of a verbal set to be zero in a free profinite group of finite rank, from which we deduce that certain existentially definable sets in free profinite groups of finite rank can only have measure zero or one. Finally, we compute the measure of definable sets in torsion-free finitely generated abelian groups.

It is well-known that any Coxeter group $W$ is a meet-semilattice with respect to weak order. Furthermore, when $W$ is finite, it is a lattice. In the first part of this paper, we give a recursive algorithm that computes joins in weak order for arbitrary finite $W$. In the second part, we complete the proof of a conjecture of Dyer expressing the inversion set of a join in terms of the union of inversion sets of each factor, under the assumption that $W$ is finite. Some proofs were developed in collaboration with Claude Opus 5.

Motivated by a question of Regev arising from his improved quantum factoring algorithm, we study how many small primes are needed to generate the group $({\mathbb Z}/q{\mathbb Z})^\times$ when each prime may be used with exponent only $0$ or $1$. We prove that, for every fixed $\varepsilon>0$ and $A>0$, there is an absolute constant $C_*$ and a set of at most $(\log Q)^{1+\varepsilon}$ primes, all at most $(\log Q)^{C_*(A+1)}$, such that for all but $O(Q(\log Q)^{-A})$ (with the implied constant depending only on $\varepsilon$ and $A$) integers $q\leq Q$, every element of $({\mathbb Z}/q{\mathbb Z})^\times$ is a product of a subset of these primes modulo $q$. The exponent $1+\varepsilon$ in the number of primes is best possible up to the arbitrary $\varepsilon$ in the exponent.

We show that for any countable group, the set of maximal subgroups is either countable or has cardinality $2^{\aleph _0}$.

We introduce collision rank and show that groups of finite collision rank are good in the sense of topological $4$-manifold theory. Finitely generated groups of collision rank two are shown to exist outside the class of subexponentially amenable groups. Such examples are given by commutator subgroups of topological full groups of Sturmian systems, and include continuum many pairwise nonisomorphic infinite simple amenable groups.

We prove various properties of the universal valued Abelian groups $\mathbb{G}_r(N)$ constructed by Niemiec, for $r\in\{1,\infty\}$ and $N\in\{0,2,3,\ldots\}$. First we show that the completion of $\Gamma_0=\bigoplus_{n\in\mathbb{N}}\mathbb{Q}/\mathbb{Z}$ (for $N=0$) or $\Gamma_N=\bigoplus_{n\in\mathbb{N}}\mathbb{Z}/N\mathbb{Z}$ (for $N\geq 2$) with respect to a generic invariant metric, bounded by $1$ when $r=1$, is isometrically group-isomorphic to $\mathbb{G}_r(N)$, recovering a result of Doucha in the case when $r=\infty $ and $N=0$. This confirms an expectation of Doucha for the group $\Gamma_N$. We combine this genericity result with a criterion of Melleray and Tsankov concerning the extreme amenability of the generic completion of a countable group to obtain the extreme amenability of $\mathbb{G}_r(N)$. We then establish the strictly stronger properties that these groups are Lévy and strongly exotic. We conclude by showing that $\mathbb{G}_r(0)$ is also monothetic, from which we obtain the existence of a monothetic group structure on the Urysohn sphere, giving a bounded version of a result by Cameron and Vershik and answering a question of Niemiec.

We define product subcomplexes and intersection complexes for compact nonpositively curved square complexes and CAT(0) square complexes. Product subcomplexes are defined as equivalence classes of local isometries from products of graphs with embedded coordinate fibers. Using their factorizations, we give a common definition of the intersection complex, extending the constructions in S. Oh's paper "Quasi-isometry invariants of weakly special square complexes" (Topology and its Applications 307 (2022)). Cubical automorphisms induce actions on these complexes. We define morphisms by comparing compatible elevations of core-map labels and their factorizations along face chains. For a compact two-sided weakly special square complex, we prove that its intersection complex is canonically isomorphic to the deck quotient of the intersection complex of its universal cover. Under a simplicity hypothesis, this recovers the image-defined reduced intersection complex in that paper, and product-base groups are the full simplex stabilizers. We also prove that quasi-isometries between universal covers of compact two-sided weakly special square complexes induce isomorphisms of their intersection complexes in this sense, and give finite counterexamples explaining why images alone do not determine the quotient in general.

For every even prime power $q>2$, we explicitly construct a rank five string C-group representation of the Chevalley simple group $G_2(q)$ using its natural seven-dimensional matrix representation. We also give a census for the string C-group representations of $G_2(q)$ of ranks four and above for every prime power $3 \le q \le 11$; providing computational evidence that rank five string C-group representations of $G_2(q)$ are rare and that this construction is exceptional among the currently known examples.

We study fixed point theorems for actions of lattices of semisimple groups. They are deduced from vanishing results for the group cohomology of $L^p$-representations. We show that for lattices in simple groups of higher rank, the cohomology with $L^p$-coefficients vanishes below the rank whenever there are no invariant vectors. As a corollary of the vanishing for $L^1$-coefficients, we obtain that every action on an acyclic simplicial complex of dimension lower than the rank has a finite orbit. This in particular proves a conjecture by Farb. The $L^p$-vanishing below the rank proves a conjecture by Gromov regarding $L^p$-cohomology of symmetric spaces.

We prove a formula for Reidemeister numbers of virtually polycyclic groups that generalises known formulas for finite groups and torsion-free virtually polycyclic groups. Based on this formula, we introduce a new invariant which we call the tamed Reidemeister number, and prove that its associated zeta functions are rational on virtually polycyclic groups. We also show that this implies rationality of (ordinary) Reidemeister zeta functions, as well as rationality of Nielsen zeta functions on compact infra-solvmanifolds.

In this short paper, we prove the bi-exactness of all wreath-like product groups $G\in\mathcal{WR}(A,B\curvearrowright I)$, whenever $A$ is amenable, $B$ is bi-exact, and the action $B\curvearrowright I$ has amenable stabilizers. As applications, we obtain solidity for the associated group von Neumann algebras and, by combining our result with existing rigidity theorems, new examples exhibiting the McDuff superrigidity.

A description of the endomorphisms of Zappa-Szép products of two groups as a group of $2\times 2$ matrices of maps is discussed in \cite{detsem}. Using this description, we have studied the concept of determinant for the endomorphisms of Zappa-Szép products of two groups. A characterization of the invertible endomorphisms is given with the help of the tools developed using the determinants. The determinants are also computed for the alternating group $A_{5}$.

The purpose of this paper is to prove that every finite simple group of type $\mathsf B$ or $\mathsf C$ satisfies the inductive blockwise Alperin weight condition at every prime $\ell$ dividing its order. For ${\rm PSp}_{2n}(q)$, where $q$ is odd and $n\geq3$, a permutation lattice argument based on Conlon's induction theorem removes the unitriangularity assumption from earlier results on the inductive condition at odd nondefining primes. At $\ell=2$ in odd defining characteristic, errors and omissions in the underlying classification of radical $2$-subgroups affect an earlier parametrisation of weights for arbitrary blocks. We verify the required weight parametrisation for principal blocks and use Jordan reduction to establish the inductive condition for all blocks. At odd $\ell$ in even defining characteristic and rank at least four, we prove the inductive condition using generic weights and Jordan reduction, without assuming unitriangularity. For $\Omega_{2n+1}(q)$, where $q$ is odd and $n\geq3$, we remove the unitriangularity assumption at odd nondefining primes using Conlon's induction theorem. At $\ell=2$, we prove the stabiliser and extension property for Brauer characters of $\operatorname{Spin}_{2n+1}(q)$ that was assumed in earlier work on the inductive condition. We also give proofs of the inductive condition for the remaining sporadic groups $J_4$, $Fi'_{24}$, the Baby Monster and the Monster, without making a priority claim. Together with the previously established cases, these results imply that the blockwise Alperin weight conjecture holds at $\ell$ for every finite group each of whose nonabelian simple sections of order divisible by $\ell$ is of type $\mathsf B$, of type $\mathsf C$, or sporadic.

We consider relatively prime integer numbers $m$ and $n$ such that each group of order $mn$ has a normal subgroup of order $m$. We prove that any skew brace of size $mn$ has an ideal of size $m$ and, if the subgroup of order $m$ of its additive group is abelian, a subbrace of size $n$. We give a method to classify skew braces of size $mn$, such that the subgroup of order $m$ of their additive group is abelian, from the classification of braces of size $m$ and the classification of skew braces of size $n$. We apply this result to determine all skew braces of size $p^2q^2$ of non-abelian type, for $p$ and $q$ odd primes, $p<q$, such that their ideal of order $q^2$ is cyclic.

Finite simple skew braces with non-abelian additive group remain largely unexplored. In particular, to the best of our knowledge, no finite simple skew brace with non-abelian nilpotent additive group had previously been constructed. We introduce an inverse Baer deformation which produces skew braces of additive nilpotency class at most two from braces of abelian type. Starting from a finite left brace $C=(A,\oplus,\circ)$ of odd order and a suitable biadditive alternating map $\kappa:A\times A\to A$, we define \[ x+y=x\oplus y\oplus\frac12\kappa(x,y). \] The resulting skew brace $\IB_\kappa(C)$ satisfies $\Br(\IB_\kappa(C))=C$ and $[x,y]_+=\kappa(x,y)$, and simplicity passes from $C$ to $\IB_\kappa(C)$. Applying this construction to simple left braces arising from asymmetric products, for every pair of odd primes $p,q$ with $p\mid(q-1)$ and every $m\ge1$ we construct a finite simple skew brace $X_{p,q}^{(m)}$ of order $p^{2m(q-1)+1}q$ such that \[ (X_{p,q}^{(m)},+)\cong E_{p,q}^{(m)}\times C_q, \] where $E_{p,q}^{(m)}$ is an extraspecial $p$-group of order $p^{2m(q-1)+1}$ and exponent $p$. Thus every fixed admissible pair $(p,q)$ yields infinitely many pairwise non-isomorphic finite simple skew braces with non-abelian nilpotent additive group of class two.

Let $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. This answers a conjecture by Erdmann, Klász and Marczinzik. We use this to show that the Hochschild cohomology of $KG$ is nonzero in every nonnegative degree whenever the characteristic of $K$ divides $|G|$.

We show that for $n \geq 6$ each element of the commutator subgroup in the symmetric group $\mathfrak{S}_n$ resp. in the signed symmetric group $(\mathbb{Z}/2\mathbb{Z})^n\rtimes\mathfrak{S}_n$ is the commutator of two $n$-cycles resp. the commutator of two $n$-cycles with a negative sign product; with one exception. If $n \equiv 2$ $\mathrm{mod}~4$, the element $-\mathrm{id}$ of $(\mathbb{Z}/2\mathbb{Z})^n\rtimes\mathfrak{S}_n$ is not such a commutator. In the language of Coxeter groups, this yields a description of commutators of Coxeter elements in types $A$ and $B$.

For an additively idempotent semiring (ai-semiring) $S$, let $S^0$ denote the ai-semiring obtained from $S$ by adjoining a new element $0$. In this paper, we develop an approach to investigate the interval $[\mathsf{V}(S), \mathsf{V}(S^0)]$ of ai-semiring varieties between the variety generated by $S$ and that generated by $S^0$. We establish a general sufficient condition under which this interval has the cardinality of the continuum. This is applied in particular to $[\mathsf{V}(S_7), \mathsf{V}(S_7^0)]$, where $S_7$ is a $3$-element ai-semiring and is a nonfinitely based algebra of the smallest possible order, thereby resolving an open problem proposed by Jackson, Ren, and Zhao (J. Algebra 611 (2022), 211–245). The same conclusion holds for $[\mathsf{V}(B_2^1), \mathsf{V}((B_2^1)^0)]$, where $B_2^1$ is the ai-semiring whose multiplicative reduct is the $6$-element Brandt semigroup. We also present a sufficient condition for the nonfinite basis property in ai-semiring varieties. As a corollary, we obtain a new proof of Dolinka's theorem (Internat. J. Algebra Comput. 17 (2007), no.~8, 1537–1551) that the $7$-element ai-semiring $(B_2^1)^0$ has no finite basis for its identities.

We prove that the first-order theory of nonabelian free groups does not admit a model companion.

Automorphism groups are among the most natural structural invariants of an algebra, and their explicit determination is a basic problem in the structure theory of Leibniz algebras. In a series of earlier papers, automorphism groups were obtained for several classes of low-dimensional, one-generated, nilpotent, and non-nilpotent Leibniz algebras. In the present paper we complete this picture for three-dimensional non-Lie left Leibniz algebras over arbitrary fields. We use a refined organization of the three-dimensional classification into sixteen isomorphism types, including parameterized families. Previously known cases are not recomputed; instead, they are incorporated by reference, with changes of basis recorded when necessary. For the remaining types we determine the automorphism groups explicitly through a direct analysis of the automorphism conditions. Particular attention is paid to exceptional parameter values and to characteristic two. A final table summarizes the automorphism group of every type in the classification.

Let \(R\) be any unital ring equipped with a Sylvester matrix rank function, and let \(Q\) be the rank completion of the matrix system defined by block-diagonal embeddings along a factor sequence. We prove that both \(\GL(Q)\) and \((Q,+)\) are exotic: every strongly continuous unitary representation is trivial. We also prove that \(\GL(Q)\) is Bourbaki bounded and admits no nonzero escape function. For every \(\varepsilon>0\), let \(B_\varepsilon\) be the open ball centered at the identity. The least positive integer \(N\) such that \(\GL(Q)=B_\varepsilon^N\) is \(\lfloor\varepsilon^{-1}\rfloor+1\). Neither regularity nor irreducibility is required. Our proof of exoticness uses a uniform Kazhdan estimate for additive integer matrix groups together with rank-one matrix perturbations, rather than the character-theoretic and continuous ring methods used previously. The geometric conclusions follow from factorizations using diagonal corner subgroups.

A random group in the square model $\mathcal{Q}(n,d)$ is given by a presentation with $n$ generators and a uniformly random set of $(2n-1)^{4d}$ cyclically reduced relators of length 4 over them. We prove that if $d>\frac{5}{12}$, then a random group in $\mathcal{Q}(n,d)$ has Kazhdan's property (T) asymptotically almost surely (a.a.s.). This provides a new construction of infinite word-hyperbolic Kazhdan groups when $d\in\left(\frac{5}{12}, \frac{1}{2}\right)$. Moreover, also for $d>\frac{5}{12}$, we show that a random group in the square model a.a.s. has property $(\textrm{F}L^p)$ for an increasing range of values of $p$, and property $(\textrm{F}_X)$ for any fixed uniformly curved Banach space $X$. These results lead to bounds on the conformal dimension of the boundary of a random group in the square model. In the process we devise a method of applying the spectral link criteria for property (T), and other fixed point properties, to the groups presented by relations of length 4. In order to use it for the square model, we verify the required spectral gap hypothesis by the trace method, in the spirit of the work of Broder-Shamir.

We establish the Margulis-Soifer dichotomy for one-relator groups: every one-relator group is either virtually solvable or has a maximal subgroup of infinite index. We also present examples of one-relator groups with and without free maximal subgroups of infinite index, as well as examples that possess both free and non-free infinite index maximal subgroups. Triviality of the Frattini subgroup is also shown for all non-solvable one-relator groups. We close the paper with a short list of questions.

We show that the $ \mathrm{K}_1 $-functor modeled on simple reductive groups over semilocal rings is solvable if the isotropic rank is at least $ 2 $ and that the Tits index is neither $ {}^{2} \mathsf{E}_{6, 2}^{16'} $ nor $ \mathsf{E}_{8, 2}^{78} $. For these two Tits indices the result is already known, but assuming that the base ring contains a field. Our result implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.

We prove that a Hall $\pi$-subgroup $H$ of a finite group $G$ is normal if and only if every irreducible constituent of the permutation character $1_H^G$ has $\pi'$-degree.