Papers

We investigate Baer's realization problem for almost free abelian groups, focusing on the extent to which rings can be represented as endomorphism rings under strong freeness conditions. Building on earlier work that relied on additional set-theoretic principles such as the diamond or strong black boxes, we develop new methods that significantly weaken these assumptions. The main result is obtained in ZFC (assuming a mild cardinal arithmetic configuration): for a strong limit singular cardinal $\mu$ with $\mu^+ < 2^\mu < 2^{\mu^+}$, and for a wide class of cotorsion-free rings, we construct $\mu^+$-free modules whose endomorphism rings are isomorphic to the given ring. This provides a substantial partial solution to a problem of Göbel and Trlifaj. The key innovation is the integration of $\kappa$-frame constructions with Shelah's Super Black Box, enabling a delicate diagonalization that eliminates nontrivial endomorphisms while preserving high degrees of freeness.

In [Finite groups with a small number of cyclic subgroups. Czechoslovak Mathematical Journal 75.3 (2025): 839-851], the authors classified all the groups having 13 cyclic subgroups. In this paper, we give a new proof for the classification of all finite groups having exactly 13 cyclic subgroups.

We show that there is a Hadamard space in which the closed convex hull of a set containing merely three points is noncompact, by providing an AI-generated construction. The seemingly innocuous problem of finding a compact or finite set with this property, or showing that none exists, was recorded by Gromov and periodically revisited by researchers in metric geometry, functional analysis, fixed-point theory, and geometric group theory. Nevertheless, it remained unanswered for over three decades. The construction itself is based on building an inductive sequence of CAT(1) metric graphs and taking the Euclidean cone over the limiting space. We discuss the history of this problem, the main ideas and details of the construction, and its relationship with existing techniques.

For an action of an inverse semigroup $S$ on a locally compact Hausdorff space $X$, we construct an étale groupoid $S\ltimes X^\mathrm{KS}$, called the KS-groupoid associated with the action. This construction may be viewed as a dynamical analogue of the construction of universal groupoids for inverse semigroups. We provide a dynamical version of Paterson's theorem which states that the full and reduced crossed products (in the sense of Khoshkam and Skandalis) of $C_0(X)$ are canonically isomorphic to the full and reduced groupoid C*-algebras of $S\ltimes X^\mathrm{KS}$, respectively. We further investigate some topological properties, functoriality and universality of KS-groupoids.

We study Gallai decomposition for groups equipped with two-sided invariant partial orders. The key algebraic step extends to arbitrary binary relations compatible with the group operation: if all left and right translations preserve a binary relation $\rho$, then every least strong module $S_{\rho}(\e,g)$, $g\ne\e$, is a subgroup. For a partial order this subgroup is convex. Thus the robust modules through the identity of an ordered group form a canonical chain of convex subgroups, with each canonical factor $H/H^-$ prime, totally ordered, or equality-ordered. We characterize exactly the subgroups that are modules, show that they form a complete sublattice of the subgroup lattice, establish overlap and inheritance results for arbitrary subgroups, and prove compatibility with quotients by normal strong subgroups. For $N$-free ordered groups the prime factors disappear. Using the robust-module decomposition of cographs, we characterize all two-sided invariant $N$-free partial orders by reduced admissible two-coloured subgroup chains, with totally ordered and equality-ordered canonical factors; the order is determined by the first nontrivial factor of each element. We also determine how the canonical decomposition restricts to arbitrary subgroups, characterize finite width and prove width divisibility for subgroups, and show that every reduced two-coloured chain is realized by an $N$-free ordered abelian group whose canonical factors are isomorphic to $\mathbb Z$.

Investigating the descendant tree T(Delta) of 3-groups G of maximal nilpotency-class, with root Delta=SmallGroup(27,3), we show for the first time that an entropy-value H(P_r)=sum P_r(G)|log(P_r(G))| can also be determined for the real probability-distribution P_r(G) of all terminal vertices outside of the mainline on this infinite digraph. These vertices arise as automorphism groups Gal(F(3,infinity,K)/K) of maximal unramified pro-3-extensions F(3,infinity,K) of real-quadratic number fields K=Q(d^1/2), d>0. For the groups G of another descendant tree T(N), with root N=SmallGroup(729,45) and unbounded coclass, which has been studied by L. Bartholdi and M. R. Bush, we compare the entropy H(P_r) of the real measure P_r(G) with the entropy H(P_i) of the imaginary measure P_i(G).

We give a criterion for property $P_{\text{naive}}$ for subgroups of orientable big mapping class groups and apply it to mapping class groups of non-orientable infinite type surfaces through the orientation double cover.

We record two results on additive diameters of group representations. First we give a corrected version of Proposition 5.3 from the first arXiv version of arXiv:2504.07573. Let $0<\varepsilon<1/3$, let $n>9/\varepsilon^2$, and let $U\leq\mathfrak{sl}_n(\mathbb{C})$ with $\dim U>\varepsilon n^2$. Then $\operatorname{diam}^{\mathrm{SL}_n(\mathbb{C})}_{+}(\mathfrak{sl}_n(\mathbb{C}),U)\leq 32/\varepsilon+8$. The proof uses an averaging argument for the action of the symmetric group on the off-diagonal matrix positions. This argument was subsequently developed further in arXiv:2609.03882. We also prove the right-hand inequality in Question 6.7 of arXiv:2504.07573. If $G$ is a complex algebraic group, $V$ is a finite-dimensional $G$-module, $U\leq V$, and $\mathfrak g=\operatorname{Lie}(G)$, then $\operatorname{diam}^{G}_{+}(V,U)\leq\operatorname{diam}^{\mathfrak g}_{+}(V,U)$. No irreducibility or connectedness assumption is needed.

We formulate the generalized hidden shift problem over arbitrary finite groups and obtain upper and lower bounds for the sample complexity in terms of group theoretic data.

In earlier work, the authors gave a construction and description of an amalgamated free product of filtered Lie algebras within a fixed nilpotency class, based on an intricate induction rooted in the work of Maier and of Higman. In this paper, the authors give a new description of this amalgam, adopting the viewpoint and methods from the work of A. I. Shirshov on amalgamation of Lie algebras, substantially simplifying their previous approach. This new description yields both group-theoretic and descriptive set-theoretic applications. A $c$-nilpotent group is called UL-equivalent if its lower and upper central series coincide. We prove that every finite $c$-nilpotent group of prime exponent $p$ with $p>c$ embeds in a finite UL-equivalent $c$-nilpotent group of exponent $p$. This recovers a result due to Ivanov and Majcher that the Polish space of enumerated $c$-nilpotent groups of exponent $p>c$ has a comeager orbit. Our result also has the following consequences. First, the class of finite $c$-nilpotent groups of exponent $p>c$ has the cofinal amalgamation property (answering a question of Ivanov and Majcher, who showed that it has the weak amalgamation property). Second, the reduct of the Fraïssé limit of $c$-Lazard groups of exponent $p>c$ to the group language is generic in the space of enumerated groups. Finally, we prove analogues of these results also for torsion-free $c$-nilpotent groups.

By a reflection group we mean an abstract group generated by nonidentity involutions. Knill proved that every reflection group of cardinality at most the continuum is natural, in the sense that some metric determines its group structure up to isomorphism among group structures whose right translations are isometries, and asked whether the cardinality hypothesis can be removed. We answer this question affirmatively. More precisely, every reflection group G admits a right invariant metric taking at most seven values for which the full isometry group is the right-regular action of G. The proof replaces distinct numerical labels on generators by a finite-valued encoding of a rigid graph on a well-ordered irredundant generating set.

We construct a simple expansion of the theory of nonabelian free groups in which the group admits a global type with bounded left-translation orbit but is not definably amenable. This extends the construction of Chernikov, Hrushovski, Kruckman, Krupiński, Moconja, Pillay, and Ramsey, and gives a counterexample to Petrykowski's conjecture.

A tubular group has a graph of groups decomposition with $\mathbb{Z}^2$ vertex groups and $\mathbb{Z}$ edge groups. This paper gives a classification of one vertex one loop tubular groups $G_{(k,\ell),(m,n)} = \langle a,b,t: [a,b]=1, t a^m b^n t^{-1} = a^k b^\ell \rangle$ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero “intersection number" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.

Let $C$ be an $[n,k]_q$ maximum distance separable (MDS) code with $4\le k\le q-3$, and suppose that it has a $2$-transitive permutation automorphism group. In this paper we show that $n\le q+1$, so the MDS conjecture holds for this class of codes.

We investigate finite group structures that yield complex irreducible characters induced simultaneously from two or more distinct subgroups. When such a group is realized as the Galois group of a Galois extension of number fields, this phenomenon implies that multiple subfields share the same Artin $L$-function.

A Dehornoy structure is a tool used to obtain a left order in the Garside group~$G$. Arcis and Paris introduced two conditions, known as Condition~$A$ and Condition~$B$. When they are both satisfied, they ensure that~$G$ possesses a Dehornoy structure. These two conditions describes how the product in the corresponding monoid behaves with an alternating decomposition. Here, we consider the case of a spherical type Artin group~$G$. We prove that Condition A almost always holds, but Condition~$B$ is rarely satisfied.

We prove the codegree isomorphism conjecture for finite simple groups. Thus every nonabelian finite simple group is determined, among finite groups, by its set of irreducible character codegrees.

We prove that every one-relator group is sofic, answering a question of Nate Brown. More generally, every group admitting a reducible presentation without proper powers is sofic, with one proper-power relation allowed at the final reducible step. The proof starts from the endpoint-preserving edge replacements of Poulin–Wróbel. We retain the free-group word carried by each terminal replacement and use it to define an \(F(S)\)-valued cocycle repair; in its relative form, the repair preserves the previously chosen generator coordinates while controlling the new relator defect. Coinduction transports the lower cocycle through successive one-relator-product extensions, allowing the construction to be iterated along reducible presentations. For a final relator \(w^m\), cyclic translates of the repaired set amplify a zero-defect set of measure close to \(1/m\) to one of measure close to one. A cocycle criterion then converts arbitrarily small relator defect into finite permutation approximations via a treeable skew-product orbit relation.

For a finite group $G$, let $\rho^{*}(G)$ be the set of primes dividing some sub-class size of $G$, and let $\sigma^{*}(G)$ be the maximum number of distinct prime divisors of a sub-class size. Motivated by the conjugacy-class version of Huppert's $\rho$–$\sigma$ problem, we prove that \[ |\rho^{*}(G)|<2\sigma^{*}(G) \] for every finite nonnilpotent group $G$. This answers \cite[Remark~5.3]{QY} and extends to arbitrary finite nonnilpotent groups the coefficient-$2$ bound proved in \cite[Theorem~1.7]{QY} for solvable nonnilpotent groups. The coefficient $2$ is best possible. The proof does not use the classification of finite simple groups.

We prove that for any cocompact Fuchsian group the class of Medvedev degrees attained by subshifts of finite type is the class of $\Pi_1^0$ degrees. This result relies on the construction of a rigid hierarchical structure in a graph model of the hyperbolic plane, and on the fact that every such group admits a representation in $\operatorname{PSL}_2(\mathbb{R})$ such that the matrices in the image have coefficients that are computable real numbers.

In this paper we apply our work on polylogarithmic cocycles representing the Borel classes to establish that the comparison map from bounded continuous cohomology to continuous cohomology is surjective for connected semisimple Lie groups with finite centre. This proves conjectures of Dupont and Monod.

In this paper, we show that $\operatorname{SL}_2(\mathbb{F}_{13})$ and $\operatorname{SL}_2(\mathbb{F}_{19})$ are Galois groups of totally real extensions of $\mathbb{Q}$. For each of these primes $\ell$, we find a polynomial of degree $\ell+1$ whose splitting field has Galois group $\operatorname{PSL}_2(\mathbb{F}_\ell)$. These fields satisfy Böge's criterion and the associated central embedding problem has a proper solution with Galois group $\operatorname{SL}_2(\mathbb{F}_\ell)$. One can choose the resulting fields totally real via a quadratic twist. The degree $14$ polynomial is found through a genus one Hurwitz family of degree 14 covers. The degree 20 polynomial is found using Yang's modular equations for the Shimura curve $X_6^*(1)$.

For any link, we define a monodromy map from the motion group of the link to the group of automorphisms of the link's Khovanov homology. The image of this map is the unoriented monodromy group of the link. This map allows us to convert results concerning motion groups of links into ones about their Khovanov monodromy. In particular, we use a characterisation of the motion groups of split links to write their unoriented monodromy groups as an explicit semidirect product in terms of their unsplit pieces. We give some demonstrative examples, computing the unoriented monodromy groups of unlinks, Hopf links, and split links composed of pieces thereof.

A second maximal subgroup of a finite group $G$ is a subgroup $H$ that is maximal in every maximal overgroup of $H$ in $G$. We prove that every maximal subgroup of a prime-index subgroup of a finite group is a second maximal subgroup.

Let a group $G$ act linearly on a finite-dimensional complex vector space $V$. The group-additive diameter of a subspace $U \leq V$ is the least number of translates of $U$ whose sum is all of $V$. Counting dimensions, it is at least $\dim V / \dim U$. We show that when $G$ is compact and $V$ is irreducible, the diameter of every nonzero subspace is at most $\lceil (\dim V / \dim U) \ln \dim V \rceil$, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity $\mathsf{C}(V)$, the largest ratio between the diameter of any subspace and its trivial lower bound, so that $1 \leq \mathsf{C}(V) \leq 2 + \ln \dim V$, and we determine where in this range various representations lie. Every irreducible representation of $\mathrm{SL}_2(\mathbf{C})$ has $\mathsf{C}(V) = 1$. The logarithm can be genuinely present along families of symmetric and exterior powers of $\mathrm{SL}_n(\mathbf{C})$ with $n$ varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as $\mathrm{PSL}_2(\mathbf{F}_p)$. The complexity is bounded above by a constant on the conjugation representations of $\mathrm{SL}_n(\mathbf{C})$ and on the representations $\operatorname{Sym}^k \mathbf{C}^3$ of $\mathrm{SL}_3(\mathbf{C})$. On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra $\mathfrak{sl}_3(\mathbf{C})$ acting on $\operatorname{Sym}^k \mathbf{C}^3$, the monomial diameter with respect to $\operatorname{Sym}^k X$ for a plane $X \leq \mathbf{C}^3$ is optimal, while the corresponding $\mathrm{SL}_3(\mathbf{C})$ diameter is not.

In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].

On every infinite compact Hausdorff abelian group there is a complex-valued continuous function whose Fourier sums, taken over coefficients above a decreasing magnitude threshold, are unbounded almost everywhere. In fact, such functions form a dense \(G_\delta\) in the space of continuous functions with its uniform norm. Moreover, a divergent example may be chosen with Fourier support contained in any prescribed infinite subgroup of the dual, while avoiding any prescribed finite set of frequencies. No metrizability or zero-dimensionality is assumed. The proof combines Körner's finite circle and Walsh lemmas with a finite construction on products of odd cyclic groups. A trichotomy for the dual group, exact quotient transfer, and cyclic sampling give finite examples on every group under consideration.

Let $F$ be the fraction field of a discrete valuation ring. We give an explicit matrix in $\mathrm{SL}_2(F[X^{\pm1},Y^{\pm1}])\setminus \mathrm{E}_2(F[X^{\pm1},Y^{\pm1}])$. The proof of non-elementarity is based on techniques developed by Peter Abramenko in his preprint arXiv:0808.1095v1.

(This is a reprint of the thesis published in the Transactions of the Conference Devoted to the 90th Anniversary Of Alexei A. Lyapunov, Novosibirsk, October 8-12, 2001, pages 323-327). The number of ways for partial $n$-quasigroups of order $4$ to be extended to $n$-quasigroups is investigated. Upper bounds on the number of $n$-quasigroups of order $4$ are obtained. The asymptotic $3^{n+1}2^{2^n+1}$ of that number is established.

We give a complete classification of the automorphisms of twisted Chevalley groups of types ${}^2A_3$ and ${}^2A_4$, and of their elementary subgroups, over commutative local rings $R$ containing $1/2$ and equipped with a non-trivial involution $\theta$. The classification covers all isogeny types and, together with our higher-rank result~\cite{EB&DM1}, completes the description for type ${}^2A_{\ell}$ with $\ell\geq3$.