Papers

We find a generating function for the number of ordered commuting pairs of elements in a symplectic Lie algebra over a finite field of characteristic $2$, following the analogous work done in odd characteristic in Fulman and Guralnick (arXiv:1611.05499).

We determine the connected components of the divisibility graph of several families of finite groups of Lie type in defining characteristic $2$, extending the result of Abdolghafourian, Iranmanesh, and Niemeyer (arXiv:1612.04410), who treated odd defining characteristic. We show that there is a distinguished component containing all nonidentity unipotent class sizes and that every other component is either an isolated vertex or an explicitly described two-vertex component. We also determine the isolated vertices outside the distinguished component. Additionally, we prove in Appendix A that the divisibility graph of a Frobenius group has exactly two components.

Tornehave and Yalçin proved that the tom Dieck homomorphism, which sends a virtual real representation to a unit of the Burnside ring, is surjective for any $p$-group $S$. We prove that this homomorphism, and the sign homomorphism it factors through, remain surjective when restricted to fusion-stable subgroups associated to a saturated fusion system on $S$. As a corollary, we close the main question posed by Mazza–Miller in arXiv:2508.07404 by showing that given a field $k$ of positive characteristic, the Lefschetz homomorphism from the Picard group of the bounded homotopy category of $p$-permutation modules to the unit group of its Grothendieck ring is surjective for all finite groups if and only if $k = \mathbb{F}_2$.

Let $d(G)$ denote the least size of a generating set of a finite group $G$. We prove that if $G$ has a family $\mathcal H$ of subgroups such that $d(H)\leq d$ for every $H\in\mathcal H$ and $\gcd\{\lvert G:H\rvert:H\in\mathcal H\}=1$, then $d(G)\leq d+1$. This gives an affirmative answer to Kourovka Problem 21.87. The proof reduces a minimal counterexample to a critical crown-based power with nonabelian socle. An exact crown multiplicity formula and a uniform lower bound for conditional generation give a lower bound for the number of crown factors. A subgroup containing a Sylow $2$-subgroup gives the contradictory upper bound, via a pointwise centralizer estimate for Sylow $2$-subgroups of finite simple groups.

We study the quasisymmetric classification of limit sets of Kleinian groups and obtain a characterization of those geometrically finite limit sets that are quasisymmetrically universal. This allows us to obtain a quasi-isometric classification of finitely generated Kleinian groups. We also obtain a classification of virtually topologically rigid geometrically finite Kleinian groups.

For every $n\ge 8$, we construct explicit families of generating pairs of $GL(n,\mathbb Z)$ and $SL(n,\mathbb Z)$ representing countably infinitely many Nielsen equivalence classes. These pairs arise as the homology images of explicit torsion generating pairs in $Out(F_n)$ and its index-two subgroup $SOut(F_n)$, yielding countably infinitely many Nielsen classes of generating pairs in both outer groups. Within each constructed outer family, all pairs become Nielsen equivalent after one stabilization, obtained by adjoining a trivial coordinate. We derive a relation-module obstruction to Nielsen equivalence of once-stabilized generating tuples. Evans's non-cancellation examples show that this obstruction is nontrivial and can distinguish Nielsen classes of once-stabilized generating tuples. We also record a quotient relation-module refinement for possible future applications.

We prove the following “Torus Trichotomy Theorem”: every finitely generated subgroup of $\pls$ is either virtually free, a central extension of a cocompact Fuchsian group by a finite cyclic group, or a one-ended, non-acylindrically-hyperbolic group containing $\mathbb{Z}^2$. In particular, we obtain the “free or surface” alternative for word-hyperbolic groups of PL homeomorphisms of the circle.

We prove that a Hausdorff topological group is tame if and only if it is precompact.

Given a countable discrete group equipped with a proper length function, one can construct a natural spectral triple on its reduced group C$^{\ast}$-algebra. A well-studied question in non-commutative metric geometry is whether the Connes pseudo-metric associated with such a triple recovers the weak$^{\ast}$-topology on the state space, thereby yielding a compact quantum metric space in the sense of Rieffel. While this metric property is known to hold for several classes of groups - including those of polynomial growth and word-hyperbolic groups - it was widely expected that not every word-length function induces a compact quantum metric space. Despite this, no explicit counterexample has been identified to date. In this note, we provide the first family of counterexamples by proving that for every integer $d \geq 2$ the canonical spectral triple of the Lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$, equipped with the word-length function associated with a finite symmetric generating set, fails to be a spectral metric space.

We determine the Morita equivalence classes of $2$-blocks with quaternion defect groups of arbitrary $2$-power order, thereby completing the proof of Donovan's conjecture for blocks of tame representation type.

We introduce $C$-lifting functors, which axiomatize lifting properties of non-abelian tensor and exterior products with respect to prescribed classes of group extensions. For a homomorphism $f\colon\Gamma\to G$, we associate to every $C$-lifting functor $F$ a relative quotient $F_f(G)$. This quotient maps epimorphically onto the subgroup determined by $F$ in every $f$-extension belonging to $C$. We show that the construction is functorial in $f$ and that, for an $f$-extension $p\colon\widetilde G\to G$, it gives a morphism of natural exact sequences. In degree two this yields an epimorphism $ \frac{H_2(G;\mathbb Z)}{f_*H_2(\Gamma;\mathbb Z)} \longrightarrow \ker p\cap[\widetilde G,\widetilde G]. $ Applying the construction to iterated tensor and exterior powers, we obtain relative Schur-Baer theorems for the lower central and derived series. We also compare the relative tensor and exterior squares, relate the exterior construction to the relative Schur multiplier $H_2(G,\Gamma;\mathbb Z)$, and derive applications to orderability. As a consequence, we prove that a virtual knot group is left-orderable if and only if it is circularly orderable.

Glauberman's characteristic $p$-functor $K^{\infty}$ has a number of remarkable properties. For example it controls $p$-transfer in any finite group provided $p \geq 5$. These notes are the author's attempt to understand and simplify this work.

We study two Kakeya set problems in compact semisimple Lie groups. In the first, motivated by the group structure, a Kakeya set is required to contain a left coset of every closed one-dimensional torus. For a compact semisimple Lie group $G$ of dimension $d$ and rank $r$, we determine the optimal Minkowski dimension for this problem, proving that it is exactly $(d+r)/2$. The upper bound is obtained from a Lie-theoretic construction associated with a Chevalley involution, while the lower bound combines incidence geometry with geometry of numbers. We also consider a local Kakeya set problem, closer to the classical harmonic-analytic formulation, in which one requires a fixed-length one-parameter arc in every direction. For this problem we obtain lower bound $(d+1)/2$ and $(d+2)/2$ for odd and even $r$, and upper bound $(d+r)/2$. We conjecture that the upper bound should be sharp.

Given a non-atomic standard probability space $(\Omega,\mu)$, we exhibit examples of non-amenable closed topological subgroups of $\mathrm{Aut}(\Omega,\mu)$ whose natural near-action on $(\Omega,\mu)$ is whirly. This answers a 2010 question by Pestov in the negative. The argument proceeds via constructing Gaussian near-actions from topologically faithful unitary representations of non-amenable whirly Polish groups.

For a finite group $H$, let $\nu(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $\Omega$ has a subset $\Delta\subseteq\Omega$ such that $|P:P_\Delta|\ge\nu(P_\Delta)$. We also prove that if a finite solvable group $H$ acts faithfully and completely reducibly on a finite module $V$, then some $x\in V$ satisfies $|H:H_x|\ge\nu(H_x)$. Consequently, we settle Gluck's well-known conjecture, open since 1985: every finite solvable group $G$ satisfies $|G:\mathbf{F}(G)|\le b(G)^2$, where $\mathbf{F}(G)$ is the largest normal nilpotent subgroup of $G$ and $b(G)$ is the largest degree of an irreducible complex character of $G$.

In correspondence beginning on August 8, 2026, V. S. Monakhov and I. L. Sokhor communicated the following question to the author: does every finite group have an unrefinable chain of subgroups whose successive indices are nondecreasing? We answer this question in the negative, even among soluble groups. For every odd prime p, we construct a soluble matrix group of degree five over the field with p elements. The group has order 64 times the eighth power of p and has no such chain. These counterexamples form an infinite family whose smallest member, obtained when p equals three, has order 419904. The construction is based on an example due to Kohler, and the obstruction follows from three elementary calculations of maximal subgroup indices.

A discrete group $\Gamma$ has the \emph{$C^*$–ISR property} if every unital $\Gamma$–invariant $C^*$–subalgebra $A\subseteq C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for a normal subgroup $N\triangleleft\Gamma$. We show that $\text{PSL}_n(\mathbb{Z})$ satisfies the $C^*$–ISR property for every $n\geq3$.

Coset incidence geometries are an important tool which connect group theory and geometry. While the representations of the unitriangular group and its pattern subgroups have been studied extensively, the underlying geometric structures of these groups has remained largely unexplored. In this article, we construct the natural coset geometry associated with each pattern group over a finite field $\FF_q$, and prove geometrical properties of theses structures. We show how combinatorial features of their defining closed sets directly correspond to properties of their parabolic subgroups, in particular the intersection of parabolics and normality of the Borel subgroup. Finally, we characterize the automorphisms of these geometries and examine how pattern geometries behave under change of field within the same characteristic $p$.

Large-scale formalization of advanced mathematics requires more than translating individual statements: it must reconstruct a coherent theory distributed across heterogeneous sources. This process raises four challenges: discovering implicit dependencies, correcting source defects, preserving semantic fidelity, and reconciling cross-source misalignments. We present FormaTheoria, an end-to-end, AI-assisted workflow that coordinates source acquisition, formalization, proof construction, recursive dependency discovery, independent review, and reconciliation, while preserving provenance and protecting approved declarations. A shared agent framework supports long-horizon execution through tool use, context compaction, review-gated termination, section-level source context, and dependency-aware batch parallelization. Applying FormaTheoria to major components of the Classification of Finite Simple Groups (CFSG), we construct a machine-checked Lean development extending through the Bender–Suzuki theorem and encompassing the Feit–Thompson Odd Order Theorem, Glauberman's $Z^*$ theorem, and the Brauer–Suzuki theorem. This development verifies an extensive body of deeply interdependent finite-group theory while providing a foundation for continuing the CFSG formalization. An empirical analysis of the code and recorded construction process supports the practical relevance of the identified challenges and illustrates the roles of the corresponding workflow components. Together, these results demonstrate how AI-assisted workflows can reconstruct mathematically significant formal theories from distributed literature by combining language-model agents with formal verification, structured review, and explicit dependency management.

We prove that the asymptotic cone of any non-abelian free group equipped with the bi-invariant word metric has elements of order 2.

For a number field $K$, a prime $p$ and a finite set of tame places $S$ we consider the groups $G_{K,S}$ - the Galois groups of the maximal pro-$p$ extension of $K$ unramified outside $S$. The tame Fontaine–Mazur Conjecture predicts that these groups have no nontrivial uniformly powerful pro-$p$ quotients. In this paper we develop a new approach to this problem using Bockstein spectral sequences and Lie-theoretic tools. This allows us to extend and refine an earlier method due to J.~Labute, who was only able to consider the case where $p^2\nmid N(\mathfrak{q})-1$ for each $\mathfrak{q}\in S$. Based on this we develop a method to verify the uniform Fontaine–Mazur property for many $G_{K,S}$ with $|S|=3$ arbitrary. Under mild conditions on $K$ we show that for infinitely many triples $S$, the groups $G_{K,S}$ have no nontrivial uniform quotients. We also exhibit a large class of these groups, which are infinite. Finally, we present numerical evidence indicating that the criteria developed here detect the uniform Fontaine–Mazur property with very high probability for $|S|=3$.

Let O be the ring of integers of a non-Archimedean local field with finite residue field, let p be its maximal ideal, and let O_2 = O/p^2. We show that the class equation of GL_n(O_2) depends on O only through the cardinality of its residue field: for two such rings O and O' with isomorphic finite residue fields, there is a canonical bijection between the conjugacy classes of GL_n(O_2) and of GL_n(O'_2) which preserves the size of every class. The argument constructs a section of the reduction map GL_n(O_2) -> GL_n(O_1) which is multiplicative on the centralizer of any element in its block Jordan canonical form, and uses it to transport the classification of conjugacy classes lying above a fixed class of GL_n(O_1) from one ring to the other. This note records the original argument for this result, developed in the author's 2010 doctoral thesis, which predates and is independent of two later proofs of closely related statements: the Ext-theoretic classification of similarity classes for n <= 4 by Prasad, Singla and Spallone, and the Hom-theoretic approach of Jambor and Plesken for general uniserial rings of length two. We record the centralizer-section argument here since it seems to be of independent interest and several colleagues have asked to see it in print.

We initiate the study of conjugacy languages in free inverse monoids. Motivated by the notion of conjugacy languages in groups and the study of conjugacy in semigroups, we introduce the language of shortest representatives of conjugacy classes and study it for free inverse monoids of rank at least $2$ under the natural notion of conjugacy. We show that, contrary to the free group case, where this language is regular, in a free inverse monoid it is neither context-free nor co-context-free. In the monogenic case, this language is context-free. We show that this non-context-freeness comes from elements with nontrivial conjugacy classes. We define an equivalence relation as follows: all elements with a nontrivial conjugacy class are related and elements with trivial conjugacy class are only related to themselves. We call this relation $\mathrm{UConj}$. We show that the language consisting of geodesics representing elements whose conjugacy class is trivial is context-free by providing an explicit context-free grammar generating it. For groups, we show that the language of minimal representatives of $\mathrm{UConj}$ classes is regular if the group is hyperbolic and for right-angled Artin groups with the standard generating set, it is piecewise testable. For virtually abelian groups, we show that there is a generating set for which this language is piecewise excluding and exhibit an example of a virtually abelian group admitting a generating set for which this language is not regular.

We introduce and study a notion of decomposition of a C$^*$-algebra over a Coxeter system based on Tits' definition of a $W$-distance. When such a decomposition comes with suitable conditional expectations, we build an associated Fock Hilbert module and reduced C$^*$-algebra $A^r$. This unifies constructions of Voiculescu [Voi85] and Caspers–Fima [CF17] and provides a noncommutative analogue of the situation of a discrete group $G$ acting on a building, in which case $A^r \cong C^*_r(G)$. We construct covariance C$^*$-algebras $\mathscr{C}(i)\supset A^r$ as a noncommutative analogue of the crossed product $C(\Omega)\rtimes_r G \supset C^*_r(G) $ where $\Omega$ is Caprace and Lécureux's minimal combinatorial compactification [CL11] of the locally finite building. Following the approach of Hasegawa [Has17] and Klisse [Kli25], we prove a universal property for the covariance algebras. This structural result yields different approximation properties, some of which are new even for the group case.

We prove that a graph $G$ is quasi-planar - i.e. quasi-isometric to a planar graph - if and only if it can be obtained by iterating the following two operations a bounded number of times: a) subdividing each edge into a path of bounded length, and b) taking the intersection graph of a family of connected subgraphs covering $G$. This applies both to infinite graphs, and to families of finite graphs with uniform constants. The backward implication relies on, and generalises, a deep result of Davies, partly proved independently by Chang, Conroy, Tan & Zheng, saying that every string graph is quasi-planar. The forward implication requires new ideas. As a byproduct of our proofs, we deduce that every contraction minor of a quasi-planar graph is quasi-planar. Moreover, if $G$ admits a tree-decomposition with adhesions of bounded diameter and quasi-planar induced bags, then $G$ is itself quasi-planar. Our results apply to other graph classes as well, and we offer various tools for understanding quasi-isometries as well as bi-Lipschitz equivalences between graphs.

For every nontrivial finite group, we prove that its quasirandom degree gives a polynomial lower bound on its uniform twin-width, whereas its minimum faithful complex representation degree gives a linear upper bound. For nonabelian finite simple groups, these two parameters coincide, so uniform twin-width is polynomially equivalent to quasirandomness in that class, yielding a new definition of quasirandomness in the sense of Gowers. We use the lower bound to prove that uniform twin-width is unbounded over finite groups, which helps us construct finitely presented groups with finite twin-width but infinite uniform twin-width. This answers a question of Bonnet, Geniet, Tessera and Thomasse. Finally, we determine the uniform twin-width of all three Thompson groups.

We provide a fairly simple list of conditions on a group $G$ acting acylindrically on a $\delta$-hyperbolic metric space implying that the group admits an action on a set that is $k$-sharp, transitive on $k$-sets and has the property that for any $k$-set $A$ the setwise stabilizer of $A$ acts on $A$ as prescribed by some fixed subgroup $\Theta\leq\sym_{k}$. In particular, this yields many easy examples of finitely generated and even presented split sharply $2$ and $3$-transitive actions.

A. A. Suslin and V. I. Kopeiko proved that the elementary linear, symplectic, and orthogonal groups are normal in the general linear group, symplectic group, and orthogonal group respectively. They also proved relative versions of these normality results with respect to an ideal of a ring. A. Bak, R. Basu, and R. A. Rao proved that the linear, symplectic, and orthogonal transvection groups are normal in the respective automorphism groups. These results are generalizations of the normality results by Suslin and Kopeiko in the setup of modules. In this paper, we prove stronger versions of the normality results proved by Bak, Basu, and Rao, which say that the relative transvection groups are normal in the respective automorphism groups.

Given a finite group $G$, we denote by $\psi(G)$ the sum of element orders of $G$. In this note, we determine finite groups $G$ such that $|H|-|K|$ divides $\psi(H)-\psi(K)$ for all subgroups $K\leq H\leq G$. Two weaker conditions are also proposed.

Let $G$ be a finite group and $H$ be a proper normal subgroup of $G$. In this note, we introduce two new functions $o(G,H)$ and $\psi''(G,H)$ involving the sums of element orders of $G$ and $H$. We prove that they satisfy certain inequalities, with equality if and only if $(G,H)$ is an equal order pair of a particular type.