Papers

For a finite group $G$ and a symmetric connection set $S$, we study the distance Laplacian ($D^L$) and distance signless Laplacian ($D^Q$) spectrum of the Cayley graph $\Cay(G,S)$ for $G$ a finite nilpotent group. Using the internal direct product decomposition over Sylow subgroups, we reduce every question to a single (possibly non-abelian) prime-power factor. In this setting, the required tools are furnished by ordinary representation theory rather than character theory. Based on this framework, we then derive a complete and explicit formula for every eigenvalue, with multiplicity, of the distance matrix, distance Laplacian, and distance signless Laplacian of $\Cay(G,S)$ for $G$ nilpotent: each eigenvalue is indexed by a tuple of irreducible characters of the Sylow factors. Finally, we use the explicit spectrum, together with Perron–Frobenius theory, to determine the eigenvalue-free interval of $D^L$ and $D^Q$.

For an irreducible character $\chi$ of a finite group $G$, its codegree is defined by $cod(\chi)=|G:\ker\chi|/\chi(1)$. In this paper, we characterize fnite $p$-groups whose nonlinear irreducible characters all have the same codegree.

For an oriented tame knot $K$, let $G_K$ be its group, with positive meridian $\mu_K$ and preferred longitude $\lambda_K$. We prove that the profinite completion $\widehat{G_K}$, marked by the conjugacy class of $\mu_K\lambda_K$, determines $K$ up to oriented equivalence. In fact, every isomorphism preserving this mark preserves the meridian and longitude with the same conjugator. The proof combines Xu's genuineness theorem with profinite JSJ theory and simultaneous boundary realization, and applies to satellite and composite knots. Consequently, Eisermann's colouring polynomials over all finite groups determine oriented knots.

In this paper, we study the poset QIso(G) of classes of subgroups of a finite group G having isomorphic quotients. Particularly, we study it for finite abelian groups and for some classes of non-abelian metacyclic groups, namely dihedral groups Dn, the group U2np and affine groups Aff(F) over finite field F. In the case of abelian groups G, we show that there exists a duality from QIso(G) to Iso(G). We show that cyclic groups are the only groups for which QIso(G), up to duality is isomorphic to its subgroup lattice. We also show that cyclic groups are the only nilpotent groups for which QIso(G), up to duality is isomorphic to its lattice of normal subgroups. We characterize finite 2-groups all of whose subgroups are q-self dual with QIso(G) being a chain. We also characterize all p-groups G of order at most p4 whose QIso(G) is a chain.

We construct two word metrics on the same seven-dimensional torsion-free nilpotent lattice whose asymptotic cones are isometric, but for which every quasi-isometry has multiplicative constant at least $1+\varepsilon$, for one fixed $\varepsilon>0$. We use a convex invariant of a fixed quasi-isometry $\Psi:N\to M$. The displacement cocycle of $\Psi$ and the right-invariant means on $N$ determine a compact convex set $\mathcal A(\Psi)\subseteq\operatorname{Hom}(V_N,V_M)$. If $\mathscr G_{\mathrm{pt}}(\Psi)$ denotes the first-layer maps of the graded homomorphisms that occur as blow-downs of $\Psi$ with varying basepoints, then $$ \mathcal A(\Psi)=\operatorname{conv}\mathscr G_{\mathrm{pt}}(\Psi), \qquad\operatorname{Ext}\mathcal A(\Psi)\subseteq\mathscr G_{\mathrm{pt}}(\Psi). $$ For every extreme point $L\in\mathcal A(\Psi)$, the dual map $L^*$ is also the degree-one part of an isomorphism of the full real cohomology rings. This cohomological compatibility, together with the realization of first-layer Pansu differentials by invariant means, yields the multiplicative constant gap stated above.

In 2003, Héthelyi and Külshammer proved that if $G$ is a finite solvable group and $p$ is a prime such that $p^2$ divides $|G|$, then $k(G)\geq (49p+1)/60$, and they constructed an infinite family of groups for which this bound was claimed to be attained. We point out that the groups in this family are not Frobenius groups, so that their class numbers are considerably larger than stated, and that in fact the bound is never attained. We then determine the best possible bound of this shape: we show that $k(G)\geq (49p+7)/60$ for all primes $p$, which is sharp for $p=17$, and that $k(G)\geq (49p+451)/60$ for all primes $p>2719$, which is sharp for infinitely many primes $p$. Thus the slope $49/60$ found by Héthelyi and Külshammer is best possible, but the correct additive constant is $451/60$ rather than $1/60$.

We provide a presentation of the braid group in which the generating set is the set of all positive half-twists, the relations are balanced positive relations, and every relation is conjugate to one in a finite list. We then prove that for any such presentation, its positive monoid is not isomorphic to the quasipositive braid monoid. Finally, we show that the quasipositive braid monoid has no proper Garside family. These results settle a problem of Calegari from the K3 problems list.

We prove that a tubular group is linear if and only if it is residually finite, equivalently virtually primitive. Every such group embeds in $\SL_n(\Z[u])$. For a primitive tubular decomposition with at most $m$ edge slopes at each vertex, our representation has dimension $2\max\{2,m\}$. In particular, Gersten's group embeds in $\SL_6(\Z[u])$, and this dimension is minimal.

We prove that every connected, locally finite, quasi-transitive graph which is quasi-isometric to a graph excluding a countable clique as a minor is quasi-isometric to a planar graph. Consequently, every finitely generated group with such a quasi-isometry is virtually planar, meaning that some finite-index subgroup admits a planar Cayley graph. We also show that a connected, locally finite, quasi-transitive graph admitting a regular map into a bounded-degree graph excluding some finite minor admits a regular map into a bounded-degree planar graph. If the source is accessible, the target need only exclude a countable clique. Combined with work of Mackay, Spriano, and the present author, this shows that a finitely presented group admits a regular map into a bounded-degree graph excluding a countable clique if and only if it is virtually planar.

The curve graph of a surface is the graph whose vertices are isotopy classes of essential, closed curves on the surface. Edges join vertices with disjoint representatives. The curve graph plays a central role in the theory of mapping class groups of surfaces. Gaster, Greene, and Vlamis have shown that the subgraph of the curve graph induced by curves in a single primitive homology class is uniquely and finitely colorable. In the analogy between the theory of mapping class groups of surfaces and outer automorphisms of free groups, the sphere graph of $M_r$, a connect sum of $r$ copies of $S_1 \times S_2$ plays an analogous role in the theory. Motivated by Gaster, Green, and Vlamis' result, we investigate $\mathcal{S}_v(M_r)$, the subgraph of the sphere graph of $M_r$ induced by spheres representing a single primitive homology class $[v] \in H_2(M_r;\mathbb{Z})$. We show that this graph is uniquely $r-1$ colorable. As a corollary, with $r\ge 3$ we obtain a nontrivial homomorphism from the Torelli subgroup $IO_r \le \operatorname{Out}(F_r)$ to the symmetric group on $r-1$ symbols; this parallels Gaster, Greene, and Vlamis' connection between their coloring and the Chillingworth homomorphism. In the course of our proof we find new smaller generating sets for mapping class groups of $M_r$ with spheres deleted.

We completely solve part (a) of Problem 17.112 in the Kourovka Notebook. We prove that a non-abelian finite simple group $G$ has a proper non-trivial ordinary $G$-permutable subgroup if and only if $G$ is isomorphic to ${\rm PSL_2(q)}$ where $q\geq4$ is a prime power and $q\not\equiv3\pmod4$, ${}^2B_2(2^{2a+1})$ where $a\geq1$, $J_1$, ${\rm Sp}_4(4)$, or ${\rm PSp}_4(p)$ where $p$ is prime and $p\equiv1,13,17\pmod{24}$. Moreover, we classify all proper non-trivial ordinary $G$-permutable subgroups of these groups. Every other non-abelian finite simple group $G$ has no proper non-trivial ordinary $G$-permutable subgroup.

A subgroup $A$ of a finite group $G$ is \emph{hereditarily $G$-permutable} if, for every subgroup $E$ with $A\leq E\leq G$ and every subgroup $B$ of $E$, some $G$-conjugate of $B$ permutes with $A$ in $E$. We give a new criterion for a finite group to be simple. A nontrivial finite group $G$ is simple if and only if it has no proper nontrivial hereditarily $G$-permutable subgroup. This proves the conjecture of Galt and Tyutyanov and answers Skiba's question in part (b) of Problem 17.112 in the Kourovka Notebook. Earlier work settled the conjecture for alternating groups, sporadic groups, exceptional groups of Lie type, and the groups ${\rm PSL}_2(q)$ and ${\rm PSU}_3(q)$. We prove the conjecture for all classical groups. The proof uses descent through parabolic subgroups and the classification of factorizations of finite simple groups.

Let $\Gamma<\operatorname{PSL}_2(\mathbb R)$ be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let $\rho:\Gamma\to G$ be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen–Margulis–Sullivan measures on $\rho(\Gamma)\backslash G/M$, where $M$ is the centralizer of a maximal split torus in a compatible maximal compact subgroup. For every $u$ in the interior of the limit cone of $\rho(\Gamma)$, correlations of smooth compactly supported functions along $\exp(tu)$, normalized by $t^{(\operatorname{rank}_{\mathbb R}G-1)/2}$, admit a complete asymptotic expansion in integer powers of $t^{-1}$. We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on $L^2(\rho(\Gamma)\backslash G)$ and for orbital counting in Riemannian balls. More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of $\Gamma$ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations. To prove these results, we develop an intrinsic cusp geometry on the limit set of $\rho(\Gamma)$ in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.

Let $\Rn$ be the variety of rings satisfying $x^n\eqid x$, where $n\geq2$, and let $\W$ be a variety of idempotent semirings with commutative addition. The equality $\Rn\circ\W=\Rn\vee\W$ holds, and this variety has subvariety lattice $L(\Rn)\times L(\W)$. Its subdirectly irreducible members that are not additively idempotent have exactly one nontrivial ring component, which is a finite field. Such a member is determined by this field and an idempotent semiring with a multiplicative identity; a separation condition on unary polynomial functions of the latter characterizes subdirect irreducibility. Every subvariety has a finite identity basis. A four-element semiring generates a variety with subdirectly irreducible members of unbounded cardinality. The results extend the Mal'cev product theorem of Wang and Shao from the absorption subvariety to arbitrary $\W\leq\Slp$.

Let $S$ denote a $k$-subset of $\mathbb{Z}$. Let $G$ be a (multiplicative) group of size $k$. Assume that there exists a function $\varphi\colon S \to G$ that is injective and that satisfies the following: If $x, y, z \in S$ and $z = xy$, then $\varphi(z) = \varphi(x) \varphi(y)$. In 1990, Forcade and Pollington conjectured that if $S = \{ 1, 2, \ldots, k \}$, then $G$ is abelian. This has been a long-standing conjecture highlighted by Richard K. Guy as an unsolved problem, and this problem appears to have remained open, with its open status having been noted by Caicedo et al. in 2021 [Electron. J. Combin.]. We succeed in proving the Forcade-Pollington conjecture in the affirmative and in full generality.

Given a finite-type orientable surface without boundary, outside of a few low-complexity surfaces, we show that any measurable embedding of the associated mapping class group into itself is a measure equivalence. This is a measure-theoretic analog to the result of Ivanov and McCarthy that mapping class groups are co-Hopfian.

We prove that Brandt semirings over nontrivial groups with at least two indices admit no identity basis with a bounded number of variables. A uniform family of finite witnesses has size linear in the variable bound and applies to groups of arbitrary cardinality. A commutative construction yields the corresponding result for flat extensions of groups of unbounded exponent and gives an exact finite basis criterion for arbitrary flat group extensions. Structural representations lead to a classification of finite natural inverse semirings and a single identity characterizing locally Clifford natural inverse structure among regular semirings. We also describe products, orthogonal sums, minimum generators, colored graph identities and linear variable complexity. Further results concern infinite generating algebras and inherent nonfinite basability.

A lazy cellular automaton $\tau:A^G\to A^G$ is determined by a finite neighborhood $S\subseteq G$ containing the group identity $e$, a pattern $p\in A^S$, and a writing symbol $a\in A\setminus\{p(e)\}$: its local rule is equal to the projection to $e$ except when it reads the pattern $p$, in which case it writes $a$. For a nonempty subset $T$ of the fiber $S_a:= p^{-1}(a)$, we introduce compatibility conditions on $p$ along the right translates $St$, $t\in T$. We show that, over every group, the existence of a compatible set is necessary for $\tau$ to be non-idempotent, and that it is also sufficient whenever $st = ts$ for all $t \in S_a$, $s \in S_{p(e)}$. In particular, compatibility characterizes non-idempotence over abelian groups, which reduces deciding idempotence to a search over subsets of $S_a$. Finally, for every dihedral group $D_n$ with $n\geq 4$, we construct an idempotent lazy cellular automaton that admits a compatible set, showing that the commutation hypothesis cannot simply be dropped.

In this paper, for a finite group $G$ and a positive integer $k$, we introduce the function $$\psi''_k(G)=\dfrac{\psi_k(G)}{|G|^{k+1}}, \quad \text{where} \quad \psi_k(G)=\sum_{g \in G}o(g)^k,$$ as the $k$-th generalization of the function $\psi''$, introduced by M. Tărnăuceanu. Different thresholds for cyclic, abelian, nilpotent and (super)solvable groups in terms of $k$ are obtained. It is shown that these thresholds often outperform similar thresholds derived from other parameters like $\psi'$ and $\psi''$. Moreover, considering $\psi''_k$ as a $(0,1]$-valued function, we study its injectivity, surjectivity and density.

We study the combinatorics of the intervals below quasi-Coxeter elements in the absolute order of finite Coxeter groups. In particular, we prove that these intervals are EL-shellable, except possibly for quasi-Coxeter elements of types $E_7$ and $E_8$, and we compute their zeta polynomials. We also show that a total order of the reflections yields an EL-labeling if it does so on all rank-two subintervals, which proves a conjecture of Williams for Coxeter elements. In order to obtain these results, we prove them for the infinite families, and we treat the exceptional types by computer. In addition, we study the related interval monoids, and we find a counterexample to cancellativity for every proper quasi-Coxeter element of type $D_n$ ($n\ge5$), $E_6$, $E_7$, $E_8$; in particular, these monoids do not embed in the corresponding interval groups. This gives a negative answer to a question posed by Baumeister, Holt, Neaime, and Rees.

We introduce weak $A_n$-formality for Galois cochain algebras and, for profinite groups whose mod-$p$ cohomology is concentrated in degrees at most two, characterise weak and genuine $A_n$-formality through finite embedding problems and the projective system they form. We then study $A_n$-formality for number fields. Let $K$ be a number field and let $p$ be an odd prime with $\mu_p\not\subset K$. For $p\geq 5$, and for $p=3$ under a natural local cyclotomic condition, we prove that every associated finite $A_3$-embedding problem is solvable. In contrast, for every odd $p$ with $\mu_p\not\subset K$, we show that no compatible family of finite-stage solutions exists, and hence $C^\bullet(G_K,\mathbb F_p)$ is not genuinely $A_3$-formal. The proof uses Chebotarev density, governing fields, and the Gras-Munnier criterion to obstruct the compatibility of finite-stage solutions, and it builds on previous work of Maire-Mináč-Ramakrishna-Tân. This shows that genuine $A_3$-formality detects a global compatibility obstruction for number fields invisible to every individual finite embedding problem and invisible to obstructions arising from individual strong Massey vanishing problems.

We show that a free-by-cyclic group with infinite order monodromy is finite index rigid, that is, any two isomorphic finite index subgroups have the same index.

Let $M$ be a compact connected oriented smooth manifold, of dimension $n\ge2$. Let $\mathcal T_MG$ be one of the following transformation groups: $Homeo_0(M,\mu)$, $Diff_0(M,\mu)$, $Symp_0(M, \omega)$ (in case $M$ is symplectic) and $Ham(M, \omega )$ (in case $M$ is symplectic). Denote by $\overline{H}_b^d(\mathcal T_M)$ reduced bounded cohomology of $\mathcal T_M$ in degree $d$, and by $\overline{EH}^d(\mathcal T_M)$ reduced exact bounded cohomology of $\mathcal T_M$ in degree $d$. In this paper we prove that if $n=2$ and $M$ is a closed surface $\Sigma_g$ or a disc $\mathbb D$, then for every $d\ge 2$ $$\dim(\overline{H}b^d(\mathcal T_M))=\infty.$$ Moreover, $\dim\overline{EH}^d(\mathcal T_M)=\infty$ if $\mathcal T_M$ is either $Diff_0(\Sigma_g,\mu)$ or $Ham(\Sigma_g, \omega)$, or $Ham(\mathbb{D}, \omega)$; or $g>1$. In case $n>2$, under certain conditions on $\pi_1(M)$, we prove that for every degree $d\ge 2$ $$\dim\overline{EH}^d(\mathcal T_M)=\infty.$$ In particular, these results hold for $\mathcal T_M$ when $M$ is a closed, orientable smooth manifold admitting a Riemannian metric of strictly negative sectional curvature of an arbitrary dimension.

A reciprocity between multiplicities of projective direct summands for modular group algebras has been known for decades. Recently, the first author extended it to finite-dimensional symmetric algebras. We further generalize the reciprocity theorem to arbitrary finite-dimensional algebras by establishing a tensor-hom adjunction for stable and costable categories.

We prove that topological conjugacy of elements of the Brin–Thompson group $nV$ is undecidable for every $n\geq2$. From a Turing machine $T$, the reduction produces two elements of $nV$, which are conjugate by an involution in $nV$ if $T$ halts on the empty tape. If $T$ does not halt on the empty tape, the topological dynamics of the two group elements are distinguished by their visits to minimal subsystems of the set of (germ-)aperiodic points. In these groups, and any finitely-generated supergroups inside the homeomorphism group of Cantor space, we also obtain undecidability of the Whitehead problem (the problem of being in the same automorphism orbit), and undecidability of group-theoretic conjugacy.

We establish a uniform small cancellation theorem for non-elementary torsion-free hyperbolic groups: given any prescribed small cancellation constant, finitely many fixed words suffice to produce a small cancellation element from every pair of noncommuting elements. As an application, we prove that the first-order theory of every non-elementary torsion-free hyperbolic group does not admit a model companion.

We prove that almost all finite groups are $2$-groups of nilpotency class two, when groups of order at most $x$ are counted up to isomorphism and $x\to\infty$. More precisely, uniformly for $2^m\leq x<2^{m+1}$, all but an $O(2^{-cm^2})$ proportion have order $2^m$ and class two, for some absolute constant $c>0$. For each fixed prime $p$, we give a formula for the number of groups of order $p^m$, with relative error $O(p^{-m/3})$ as $m\to\infty$. This refines the classical logarithmic estimates for the number of $p$-groups. For every fixed prime $p$ and sufficiently large $m$, we also prove that all but a $p^{-m^2/300}$ proportion of groups of order $p^m$ have central elementary abelian Frattini subgroup. For fixed $p$, the distribution of the minimum number of generators among groups of order $p^m$ is asymptotically supported on one or two adjacent values, with explicit probabilities depending on $m$ modulo three.

The Whitehead problem, which asks whether every abelian group $A$ satisfying $\mathrm{Ext}^1(A,\mathbb{Z}) = 0$ is free, is independent of ZFC. However, this problem has an analogue in condensed mathematics that can be answered affirmatively. In this note, we give a new proof that an abelian group $ A $ of size $\kappa$ is free if and only if $\underline{\mathrm{Ext}}^1(\underline{A}, \underline{\mathbb{Z}})(S_\kappa) = 0$, where $ S_\kappa $ is the Stone space of the Boolean completion of $\operatorname{Add}(\omega,\kappa)$.

We prove that the three-element flat semiring $S_7$ is strongly nonfinitely based within the class of additively idempotent semirings. More generally, let $\mathcal N$ be the subvariety of $\Var(S_7)$ defined by $x^2y\approx x^2$. For every finite ai-semiring $A$, each variety $\mathcal W$ satisfying $\mathcal N\subseteq\mathcal W\subseteq\Var(A)$ has infinite axiomatic rank.

We develop two new general frameworks for constructing finitely generated residually finite groups that possess infinite girth, satisfy no non-trivial group law, and contain no non-abelian free subgroups. While such groups are rare–with known examples including the first Grigorchuk group, Thompson's group F, and certain iterated wreath products–our approach provides a more explicit construction of such groups realized as subgroups of direct products of symmetric groups. We investigate the relationship between the girth of a group and that of its finite quotients, leading to the introduction of a "marked" girth criterion, which we establish as a necessary and sufficient condition for infinite girth within the class of residually finite groups. Furthermore, we explore the connection between girth and profinite completions and provide a congruence realization result for infinite girth in linear groups. We conclude by posing several open questions that arise naturally from our results.