Papers

We provide a complete characterization of a finite group in which every non-supersolvable maximal subgroup is a TI-subgroup, which generalizes some known related results.

We characterize the central extensions, in the sense of the categorical theory of central extensions of Janelidze and Kelly, relative to the Birkhoff subcategory of $n$-nilpotent objects of a semi-abelian and algebraically coherent category. An extension $f \colon A \to B$ with kernel $K$ turns out to be central precisely when the iterated Huq commutator $[A,[A,\dots,[A,K]\dots]]$, with $n$ occurrences of $A$, vanishes. For $n=2$ this condition reads $[A,[K,A]] = 0$, that is, $[K,A]$ is a central subobject of $A$. In the special case of groups this description was obtained by Everaert and the first author. The present proof is categorical, and applies in particular to Lie algebras, to crossed modules, to cocommutative Hopf algebras over a field, and to any category of interest.

For every countable discrete group, we construct a character for finite-dimensional unitary quasi-representations, with values in the inverse limit of the equivariant $K$-theory of its Rips complexes. The character is a complete invariant for quasi-representations up to stable equivalence for amenable groups and, more generally, for groups satisfying the Haagerup property and QR-quasidiagonality. For Haagerup groups satisfying quantitative QR-quasidiagonality, we also obtain a quantiative version of the theorems. These conclusions apply when the maximal group $C^*$-algebra is quasidiagonal and has the lifting property, including amenable groups, free groups, and closed orientable surface groups. For surface groups, rank and winding number give complete invariants for the classification. Finally, composing with the equivariant Chern character gives a delocalized cohomological obstruction to stable equivalence of a quasi-representation with genuine representations.

We construct an explicit countable group \(G_*=N(R)\rtimes A\) whose generated variety is a limit variety of groups: it has no finite identity basis, whereas every proper subvariety is finitely based. Here \(A\) is relatively free in the variety of groups of nilpotency class at most two and exponent dividing four, and \(R\) is an explicitly defined quotient of the group algebra \(\mathbb F_2[A]\). The proof establishes an ascending chain condition for coefficient spaces indexed by six ordered commutator entries and reduces every proper subvariety to a hereditarily finitely based approximation. Non-finite basability is obtained by transferring independent identities from the constructions of Gupta and Krasilnikov. The resulting variety is contained in \(\mathfrak B_4\mathfrak B_2\). This construction provides the first example of such a variety in which both the generators and a basis are explicitly given.

Let $G$ be a semisimple Lie group that is not isogenous to the isometry group of real or complex hyperbolic $n$-space. We show that the rational rank is a profinite invariant for irreducible lattices in $G$, unless $G$ has type $D_{2n}$ which forms a genuine exception. In particular, profinitely isomorphic lattices in $G$ of type $D_{2n}$ can have differing virtual cohomological dimension. As another corollary, uniformity is a profinite property for irreducible lattices in $G$ outside type $D_4$. Assuming CSP, the existence of triality symmetries gives uniform and non-uniform profinitely isomorphic lattices in type $D_4$.

Let $G$ be a finite abelian group of order $n$ and $X$ a finite $G$-set. We give a necessary and sufficient condition on $G$ for any $X$ to be determined up to isomorphism by certain combinatorial data. Specifically, if $M_H^X$ is the number of $H$-orbits in $X$, and $a_d^X$ is the number of $G$-orbits in $X$ of cardinality $d$, we characterize when equalities $M_H^{X_1} = M_H^{X_2}$ and $a_d^{X_1}=a_d^{X_2}$ for all $H \leq G$ and $d|n$, respectively, force an isomorphism $X_1 \simeq X_2$. Our main result is that this occurs if and only if $G$ is cyclic or $G$ is the direct sum of a finite cyclic group and a cyclic group of prime order. We conclude with an arithmetic application to $L$-function factorization.

The total variation profile $\mathrm{TV}_n$ of a random walk measures how far its distribution at time $n$ is from being invariant under the generators. Three lower bounds are given: an elementary bound $\mathrm{TV}_n \geq 1/\big(44\,L(n)\big)$ in terms of the speed $L(n)$; a bound by the entropy increment $H(P^{n+1}) - H(P^n)$, which is a reverse Pinsker inequality; and a bound in terms of the entropy and the isoperimetric profile. Deciding whether the entropy-increment bound always improves on the isoperimetric one is related to conjectures on the growth of the entropy and of groups. The motivation comes from the question of whether the distributions of a random walk can be transported back onto their translates at bounded cost.

Let $G$ be a finite group and let $\pi(G)$ be the set of prime divisors of $|G|$. For each $p\in \pi(G)$, choose a Sylow $p$-subgroup $G_p$ of $G$. The subgroup $\Psi(G)= \prod_{p\in \pi(G)} \Phi(G_p)_G$ is called the local Frattini subgroup of $G$. Here, for $J \leqslant G$, the symbol $J_G$ denotes the core of $J$ in $G$. The subgroup $\Psi(G)$ is a characteristic subgroup of $G$ that is contained in $\Phi(G)$, the Frattini subgroup of $G$. We prove that $\Psi(G)$ satisfies many properties similar to $\Phi(G)$. Let $J$ be a normal $p$-subgroup of $G$ with $J \leqslant \Phi(G)$, and let $S$ be a Sylow $p$-subgroup of $G$. John S. Rose proved that if $J \leqslant \mathbf{Z}(S)$, then $J \leqslant \Phi(S)$. So, in fact, $J \leqslant \Phi(S)_G \leqslant \Psi(G)$. Kaplan, Lev, Mann, and Solomon found various other conditions that would ensure $J$ is contained in $\Phi(S)$. In this article, we add a few more conditions to this list.

We study finite involutive multivalued groups associated with Schur rings. Their reduced valency is shown to equal the least common multiple of the sizes of the basic sets of the corresponding ring. We prove that the double-coset group associated with a complemented subgroup of a finite group is isomorphic to a Schur multivalued group over the complement. In particular, every coset multivalued group is a Schur multivalued group. Using affine Schur rings, we construct families of pairwise nonisomorphic commutative multivalued groups over elementary abelian groups. We obtain asymptotic bounds for the sizes of these families and prove that the proportion of double-coset groups in them tends to zero.

We establish lower bounds on the orbit cardinalities of $\text{SL}(2,\mathbb{Z})$ on the $\mathbb{F}_p$-points of the generalized Markoff surfaces given by $x^2 + y^2 + z^2 - xyz = \tau+2$. We show that outside of the reductions of a certain exceptional finite set of $\overline{\mathbb{Q}}$-points found by Dubrovin and Mazzocco, every orbit has cardinality $\ge p$. Together with a recent result of Nathaniel Kingsbury-Neuschotz, this establishes, for all but finitely many primes $p$, the classification conjectures of McCullough-Wanderley (2013), and gives a complete classification of the connected components of the Hurwitz stack of $\text{SL}(2,\mathbb{F}_p)$-covers of elliptic curves, only branched above the origin. This extends earlier work of Bourgain-Gamburd-Sarnak and the author in the Markoff case $\tau = -2$. Our method studies certain noncongruence modular curves attached to a $\text{SL}(2,\mathbb{Z})$-orbit. These modular curves parametrize elliptic curves with a $\text{PSL}(2,\mathbb{F}_p)$-cover. We use this moduli interpretation to show that the noncongruence modular curves involved have bad reduction, as a cover of the $j$-line, at some place above $p$. This implies that the degree of the cover is $\ge p$, which translates into the desired bound on orbits. Key inputs include a result of Bouw-Wewers on bad reduction of three-point $\text{PSL}(2,\mathbb{F}_p)$-covers of the projective line, and the general stable reduction theory of Henrio, Obus, Raynaud, and Wewers.

For any nonarchimedean local field $k$, we exhibit the first examples of Zariski-dense, discrete subgroups of special linear groups over $k$, which are not commensurable to any lattice in the points of a semisimple group over a local field, and contain an infinite index subgroup isomorphic to a lattice in the $k$-points of a higher rank $k$-semisimple group. We prove that any Gromov-hyperbolic double of a non-elementary free group along a cyclic subgroup embeds as a quasi-isometrically embedded subgroup of $\mathsf{SL}_5(k')\times \mathsf{SL}_5(k')$, where $k'$ is a discretely valued field. More precisely, $k'$ is the field of rational functions $k(t)$ equipped with the Gauss valuation and $k$ is any nonarchimedean local field of characteristic not equal to $2$ or $3$.

We give negative answers to two questions of Farjoun about localizations of groups. For every odd prime $p$, we construct a finite $p$-group $G$, an infinite non-nilpotent group $H$, and a local embedding $f\colon G\hookrightarrow H$. The associated localization satisfies $L_f(G)\cong H$, showing that localizations need not preserve nilpotence and that the localization map $G\to L(G)$ need not be surjective when $G$ is a finite $p$-group.

Motivated by the recent study of $\lambda$-fold near-factorizations of finite groups by Kreher, Li, and Stinson (Designs, Codes and Cryptography, 2026), we introduce and systematically investigate $(\lambda,\mu)$-fold factorizations of finite groups, with particular emphasis on the abelian case. These structures form a natural relaxation of $\lambda$-fold near-factorizations and encompass several related families that have previously been studied in different contexts. We establish the foundational theory of $(\lambda,\mu)$-fold factorizations, provide two general constructions, and prove several nonexistence results for finite abelian groups.

Minimum product-set problems ask how small a product of two subsets of prescribed cardinalities can be in a finite group. Although the problem is completely understood for finite abelian groups, exact results in the nonabelian setting are available only for particular classes. We determine the minimum product-set function $\mu_G$ for every finite metacyclic group, without assuming that the relevant cyclic extension splits. Our proof converts the extremal problem into a finite recursive arithmetic description of the feasible cardinality vectors of subset trios. The recursion combines the subgroup structure of finite metacyclic groups with the structural classification of maximal critical subset trios. The resulting characterization recovers the previously known formulas for abelian and nilpotent groups, dihedral groups, and nonabelian groups of order $pq$. It also yields metacyclic examples in which the minimum product-set size lies strictly between the classical lower and upper arithmetic bounds for finite solvable groups. For faithful split metacyclic groups, we further show that requiring one factor to be symmetric, contain the identity, and generate the whole group can force a strictly larger product set than the unrestricted minimum. The latter constraint admits a vertex-isoperimetric interpretation on connected Cayley graphs.

For any $n\geq 1$, we construct a continuum of non-conjugate actions by homeomorphisms of $\operatorname{SL}_{n+1}(\mathbb{Q})$ on the $n$-sphere $S^{n}$ that do not extend to a continuous action of $\operatorname{SL}_{n+1}(\mathbb{R})$. We do this by adapting a blow-up construction of Bowden and Mann. We motivate studying these actions and note down some related observations.

We construct a Hausdorff quasitopological abelian group that is second-countable and semiregular, with a bounded continuous function that does not factor through any regular first-countable semitopological group. Its regular and Tychonoff reflections coincide, have a countable network, and fail even factorizability through arbitrary first-countable semitopological groups. This gives negative answers to \cite[Problems~6.1, 6.2, 6.4, and~6.5]{Tk2026} of Tkachenko. The construction starts with a countable cone group whose common separation reflection is not Fréchet–Urysohn, answering \cite[Problems~6.8 and~6.9]{Tk2026} negatively. In contrast, symmetrization gives affirmative answers to \cite[Problems~6.6 and~6.7]{Tk2026} without separation assumptions. We also show that every countable semitopological group admits simultaneous factorizations of countable families through second-countable Tychonoff semitopological groups.

For a fixed prime $p$, a group is called a $p'$-group if the orders of all its elements are coprime to $p$. It is well known that in a finite $p'$-group an automorphism of order $p$ is regular if and only if it is a splitting automorphism of period $p$. In the Kourovka Notebook, V.\,D.~Mazurov asked whether these two notions coincide for periodic $p'$-groups. We answer this question in the negative by constructing a periodic group of prime period $q$ that admits a regular automorphism of order $p \neq q$ which is not splitting.

Let $G$ be a finite $2$-transitive permutation group. We determine the orbits of $G$ on the unordered triples of its point set, together with the three-point stabilizer and its image inside $S_3$, for every affine and almost simple family in the finite $2$-transitive classification. These data classify the $G$-invariant triple systems and the simple $2$-\mbox{$(v,3,\lambda)$} designs admitting $G$, and yield in a uniform way the standard objects attached to them: the incidence and block graphs (with arc-transitivity and strong regularity in the Steiner case), the constant-weight and design codes, and the Schurian coherent configurations on triples.

We study closed-set stabilizers in topological full groups of minimal effective ample groupoids with Cantor unit space. For almost finite or purely infinite groupoids, we extend the maximal stabilizer theorem of Grigorchuk and Vorobets to every subgroup containing the commutator subgroup. For a nonempty closed nowhere dense set, maximality of its stabilizer is equivalent to minimality of its stabilizer action, minimality of the regular reduction, and primeness of a canonical wide open subgroupoid. Without assuming almost finiteness or pure infiniteness, we prove spatial rigidity for stabilizers acting minimally on infinite closed nowhere dense sets. Their isomorphism types determine the ambient Kakutani equivalence class and, when Hausdorff, the effective reduction. We also realize every AF groupoid with compact unit space as the full restriction, with its relative topology, of any minimal effective ample groupoid with Cantor unit space to a closed nowhere dense subset. Combining rigidity and realization yields continuum many pairwise nonisomorphic stabilizers in every subgroup containing the commutator subgroup; these stabilizers are maximal when the ambient groupoid is almost finite or purely infinite. The realization also gives explicit AF subquotients of the ambient reduced groupoid $C^*$-algebra, including the CAR algebra, with compatibility on the canonical diagonal subalgebras.

In a previous work, we defined a quotient graph of the Bruhat-Tits tree associated to the degree two special linear group over a local field whose vertex set is in correspondence with the maximal orders in a certain number of spinor genera. This graph has been applied to the study of the structure of some arithmetic groups, their presentations and their (co)homology groups. The genera of Eichler orders, which, for instance, are related to the problem of finding a basis that simultaneously splits two vector bundles with the same generic fiber, can often be described in terms of simpler genera via quotients. Furthermore, some embedding problems of orders can be studied via quotient graphs. These results provide a geometric framework for the study of maximal orders over function fields. In this article, we extend that construction to a higher dimensional setting through the use of singular chain complexes. Then we describe the set of vertices in this quotient that correspond to split maximal orders and the topology of the corresponding region of the quotient complex. More precisely, we show that these “split vertices” fall into a finite number of quotients of apartments by suitable subgroups of the corresponding Weyl group, the symmetric group. We relate these “quotient apartments” with the cuspidal sectors defined by previous authors who studied the quotient by the group of integral points of the special linear group. Both quotients are indeed related by a ramified cover that is unramified on cusps, as proved here.

We give an explicit treatment of the third tensor power of the Clifford representation on $n$ qutrits, for $n\geq2$. This representation has five pairwise inequivalent irreducible constituents, with multiplicities $1,1,1,1,2$. The decomposition reduces the Clifford twirl of a tensor cube to four real parameters. Two of them are spectral moments of the input. The other two are cubic invariants expressed as sums of cubes of the Weyl characteristic function and the discrete Wigner function. On trace-one Hermitian matrices, the invariant polynomials of degree at most three are spanned by the constant function and four algebraically independent functions. These functions do not separate Clifford orbits. For pure inputs a single parameter suffices. We recover its range and the characterization of its maximum by stabilizer states. We bound the stabilizer fidelity, determine full separability across the three copies in the symmetric family, and recover the criterion for a complex projective $3$-design. We classify the jointly invariant quantum Boolean functions. For stabilizer testing and for testing unknown observables we give explicit bounds on copies and queries. We also give character formulas for the finite projective Clifford groups and record the complete character table for two qutrits.

We solve Brauer's Problem 33 by determining the finite $2$-groups that occur as Sylow $2$-subgroups of finite groups with a single conjugacy class of involutions. Besides the Sylow $2$-subgroups arising from the relevant non-abelian simple groups in Table 1, these are precisely the homocyclic, generalized quaternion, and Suzuki $2$-groups. The classification of finite simple groups is used in this paper.

Twisted equations in free groups are equations in which a finite group of symmetries acts on the free group by automorphisms and anti-automorphisms, such as inversion and reversal; moreover, the variables may be subject to rational constraints. Algorithms from the theory of equations in virtually free groups solve such equations in PSPACE, but they are intricate. Our main result is an elementary algorithm for quadratic twisted equations, which extends the algorithm of Comerford and Edmunds (1981) for quadratic equations in free groups to the twisted setting and to rational constraints. Besides deciding solvability in PSPACE and yielding EDT0L descriptions of solution sets, our algorithm computes, for every quadratic expression, finitely many parametric forms describing its values as freely reduced words. As an application, we study the palindrome length of elements of free groups, that is, the least number of palindromes whose product they are. Guided by the parametric forms that our algorithm computes for products of palindromes, we found, and then proved directly, a characterization of the freely reduced words of palindrome length at most $n$ in terms of patterns describing how palindromes are nested by twisted conjugations, thereby confirming a Catalan structure conjectured by the second author. Consequently, these words form a context-free language, and palindrome length is computable in cubic time as well as in LOGCFL, answering a question of Bardakov, Shpilrain, and Tolstykh (2005).

Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. We prove, uniformly over all primes $p$, that \[ \log_p f(p^m)=\frac{2}{27}m^3-\frac49m^2+O(m), \] with an absolute implied constant. This improves the classical upper bounds of Sims (Proc. Lond. Math. Soc., 1965) and Newman–Seeley and matches Higman's lower bound (Proc. Lond. Math. Soc., 1960) through the quadratic term. We also prove that \emph{the proportion of $2$-groups of nilpotency class $2$ among all groups of order at most $n$, counted up to isomorphism, tends to $1$ as $n\to\infty$.} These results answer questions~22.3 and~22.19 of Blackburn–Neumann–Venkataraman (Enumeration of Finite Groups, 2007). Our proof refines Sims' enumeration framework using graded Lie brackets, constraints on deeper commutator coordinates, and incidence and Fourier estimates for alternating forms. Normalization by changes of generating bases then determines the quadratic term in the upper bound. We also prove that all but a $p^{-\Omega(m^2)}$ fraction of groups of order $p^m$ have a central elementary abelian Frattini subgroup. Combining this concentration with the sharp enumeration bound of Bläser–Li–Qiao–Rogovskyy (arXiv, 2025) for $p$-groups of Frattini class $2$ yields the linear error term. The same presentation estimates give fixed-quotient extension bounds which, together with Pyber's prescribed-Sylow theorem and the concentration result, establish domination by $2$-groups of nilpotency class $2$.

Let $G$ be a finite simple graph on the vertex set $[n]$, and let $\Gamma(G)$ be the Cayley graph of the symmetric group $S_n$ generated by the transpositions corresponding to the edges of $G$. We study the inverse spectral problem for the assignment $G\mapsto \Gamma(G)$ through the group-algebra operator $A_G=\sum_{\{i,j\}\in E(G)}(ij)$ and its regular spectral moments $M_k(G)=\operatorname{tr}(A_G^k)$. The moments admit two all-order forms: a universal support expansion in non-induced subgraph counts of $G$, and a representation-theoretic expansion over the irreducible representations of $S_n$. We compute the first nontrivial moments through order six. In particular, the fourth moment already contains the Coxeter triangle relation and determines the signed local invariant $6N_{K_3}(G)-N_{P_3}(G)$, rather than $N_{P_3}(G)$ alone. Consequently, for triangle-free graphs the spectrum determines the second moment of the degree sequence. We also give a simplified sixth-moment formula for triangle-free graphs, showing that in this class the spectrum determines $5N_{K_{1,3}}(G)+2N_{P_4}(G)+12N_{C_4}(G)$. Further structural consequences include bipartiteness and spectral symmetry of $\Gamma(G)$, a precise connectedness test, and the identification of the standard irreducible component with the shifted graph Laplacian $mI-L_G$.

We say that a group $G$ has the $R_{\infty}$ property if the number $R(\varphi)$ of twisted conjugacy classes is infinite for any automorphism $\varphi$ of $G$. The $R_{\infty}$-nilpotency degree is the least integer $c$ such that $G/\gamma_{c+1}(G)$ has property $R_{\infty}$; in case this integer does not exist, we say that $G$ has infinite $R_\infty$-nilpotency degree. In this paper, we determine the $R_{\infty}$-nilpotency degree of some lollipop GBS groups by using different approaches, such as Sylow subgroups, normal forms and presentation of nilpotent quotients, and diagrams of exact sequences. We also compute the $R_\infty$-nilpotency degree of all 2-generated lollipop GBS groups and of all GBS groups whose associated graph is a tree.

We prove that every length-preserving automorphism of a nonspherical Artin or Coxeter group has infinite Reidemeister number. As a consequence, we establish property $R_\infty$ for large-type free-of-infinity Artin groups, infinite strongly rigid Coxeter groups, and infinite virtually abelian Coxeter groups, including all affine Coxeter groups. We also prove that the automorphism group of every infinite irreducible non-affine strongly rigid Coxeter group is acylindrically hyperbolic.

Abstract polytopes and hypertopes are thin and residually-connected incidence geometries or, equivalently, chamber systems. Their degree of regularity is understood by looking at the action of their automorphism group on the set of their chambers. When this action has only one orbit, they are called regular. In this article, we are interested in polytopes and hypertopes with exactly two orbits on their chambers. Let $I = \{1,2,\ldots,n\}$. For any $J\subsetneq I$ with $|I| \geq 3$, we show that there exists a class $2_J$ two-orbit hypertope whose diagram is any graph on $|I|$ vertices, with edges all labeled by $\infty$. In order to do so, we first establish a general framework for constructing hypertopes as quotient of chamber systems, and then exploit the machinery of acylindrically hyperbolic groups for showing that this framework can be used to create two-orbit polytopes and hypertopes of class $2_J$. Note that all polytopes and hypertopes constructed in this paper are infinite.

We complete the proof that every compact 3-manifold group is linear. Specifically, we construct a faithful representation in GL5(Q[u]) for a finite-index subgroup of every closed non-geometric graph-manifold group.

In the context of Fell bundles over discrete groups, we introduce and analyze suitable notions of direct sum and tensor product of actions of Fell bundles on Hilbert bundles. For $C^*$-dynamical systems, where actions arise from equivariant representations, we discuss how the tensor product of two such actions is related to the action associated with the tensor product of the underlying equivariant representations. Likewise, since any action gives rise to a $C^*$-correspondence between the full cross-sectional $C^*$-algebras of the involved Fell bundles, we examine the connection between the $C^*$-correspondence associated with the tensor product of two given actions and the tensor product of the associated $C^*$-correspondences. We also discuss how the tensor product of actions naturally appears in Rieffel-like induction processes. Finally we present related results on the Fourier-Stieltjes and Fourier algebras of a Fell bundle.