Papers

We introduce the $\textit{expanded core graph}$, which records minimal unbounded domains and their axial directions, for two classes of hierarchically hyperbolic group structures modeled on compact special groups and mapping class groups. Our main structural result shows that every embedding of a right-angled Artin group, after replacing its standard generators by positive powers, factors through an intermediate RAAG generated by suitably supported axial elements; for the class modeled on compact special groups, this intermediate RAAG is quasi-isometrically embedded. We show that its extension graph embeds into the expanded core graph. This yields a Kim–Koberda-type obstruction to RAAG embeddings and a complete embedding criterion when the rank is at most two. For the standard HHG structure on a mapping class group, the expanded core graph is the disjointness graph of essential curves, while for natural rich-family structures on a RAAG it recovers the extension graph. We also establish permanence results under finite direct products and the standard relatively hyperbolic construction.

Let G = UT(4; p) be the group of all 4 ? 4 unitriangular matrices over the fi?nite fi?eld Fp, where p is a prime. Using the six-parameter form of the elements of G, we describe the commutativity relation explicitly and use it to analyse the struc- ture of the graph. We prove that the reduced commuting graph is connected and has diameter 3. We also determine the size of its maximal cliques, chromatic number, independence number, per- fectness, etc. by decomposing the graph into cosets, layers, and direction parts.

Given a finite group $G$ and a factorization of its order, $|G|= a_1 a_2,$ we show that a sufficient condition for there to exist subsets $A_1,\,A_2\subseteq G$ such that $|A_1| = a_1,$ $|A_2| = a_2,$ and $G = A_1\,A_2,$ is that there exist a chain of subgroups $\{e\} = G_0 < \dots < G_n = G$ such that $a_1$ is the product of some subfamily of the indices $|G_i:G_{i-1}|$ (and hence $a_2$ is the product of the complementary subfamily).

For a finite group $G$, denote by $k'(G)$ and $L(G)$ the number of conjugacy classes of subgroups and the subgroup lattice of $G$, respectively. Let $d'(G)=\frac{k'(G)}{|L(G)|}$ and $d^*(G)$ be the minimum value of $d'(S)$, when $S$ runs through all sections of $G$. In this paper we deduce some criteria on the nature of $G$. We show that if $d^*(G)>\frac{9}{59}$, then $G$ is solvable, while if $d^*(G)>\frac{1}{2}$, then $G$ is a supersolvable group. The last criterion is also valid when replacing “supersolvable" with “CLT".

Let $G$ be a torsion-free one-ended hyperbolic group and let $\phi\in\operatorname{Aut} (G)$. We prove that the mapping torus, or suspension, $$ M:=G_{\phi}=G\rtimes_\phi \mathbb{Z} $$ has solvable conjugacy problem. This builds on the pioneering work of Préaux who solved the conjugacy problem for all (geometrisable) three-manifolds. We view $M$ as a generalisation of a fibred three-manifold. Our proof begins with the canonical JSJ tree of $G$, whose suspension gives a graph-of-groups decomposition of $ M $. We refine each suspended QH vertex using a Nielsen–Thurston reduction system for its induced monodromy, taking account of non-orientable surfaces and orientation-reversing monodromy. Equivalently, this is a further geometric JSJ decomposition for each vertex which is a fibred three-manifold, but allowing Klein bottles as well as tori in the splitting. We then `block' pieces of this refined JSJ together according to whether they share a central element, glued along an elementary vertex, by a sequence of folding operations. This new splitting is cocompact and acylindrical; in particular certain local `solution sets' turn out to be rational subsets of virtually abelian subgroups - the virtually abelian groups in question are products of the edge groups. Our version of Préaux's graph-of-groups argument then reduces global conjugacy to effective intersection and non-emptiness for these rational sets.

Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $\mathbb{Q}$ during 1984-1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group $M_{23}$, occurs as a Galois group over $\mathbb{Q}$. In fact, we produce an explicit degree $23$ polynomial with rational coefficients whose splitting field has Galois group $M_{23}$ over $\mathbb{Q}$. To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of $M_{23}$. We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of $\mathbb{Q}(t)$ with Galois group $M_{23}$. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Costa, Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an $M_{22}$-extension of $\mathbb{Q}$.

Let $G$ be a finite group. We study the additive map on the Burnside ring of $G$ that sends an orbit $[G/H]$ to $[G/\operatorname{core}_G(H)]$, which we call the normal-core compression, and investigate its behavior on units. For a normal subgroup $N\trianglelefteq G$, we show that the cumulative $N$-core coefficient is equal to the sum of the orbit-basis coefficients of the $N$-fixed-point element over $G/N$. Consequently, for a unit this coefficient takes only the values $-1,0,1$, while the individual core coefficients are recovered by Möbius inversion on the lattice of normal subgroups. Using Yoshida's criterion, we further express the cumulative core coefficient in terms of the $N$-mark and a linear character of $G/N$. This yields a necessary and sufficient condition for the existence of a unit with nonzero $N$-core coefficient in the normal partial Burnside ring: $G/N$ must be an elementary abelian $2$-group. On the other hand, the units mapped to $1_{\Omega(G)}$ by the core compression, called core-trivial units, are characterized by the condition that all marks at core-cyclic subgroups are equal to $1$. Using primitive idempotents of the rational Burnside ring and Yoshida's criterion, we realize the group of core-trivial units as the kernel of an ${\mathbb F}_2$-linear defect map. Finally, we obtain a splitting along normal quotients and a direct-sum decomposition along direct products for this defect map. Combining the direct-product decomposition with an explicit nontrivial example for $S_4$, we show that the ranks of core-trivial unit groups are unbounded among finite groups.

We prove that every countable inner amenable group has vanishing $\ell^2$-Betti numbers in all degrees. This was previously open in every degree $n\geq2$.

We study the classifying map $c\colon BX\to K(\As(X),1)$ of a rack $X$ of finite type. Let $t = \Type(X)$. We prove that $t c_{n*}=0$ for every $n\ge2$ when $X$ is connected, and that $t^{n-1}c_{n*}=0$ on the torsion subgroup $\Tor H_n^\mathbb{R}(X)$ without any connectedness assumption. For a finite rack, under our sign conventions, the rationalized classifying map in degree $n$ is given by $(-1)^n$ times the canonical projection from the $n$-fold tensor power of the orbit module to its $n$-th exterior power. For an arbitrary rack of finite type, we determine $H_2^{\mathrm{gr}}(\As(X);\mathbb{Z}[1/t])$. We also derive low-dimensional applications to symplectic and Alexander structures.

We give infinite and finite presentations of the cactus Thompson group. This group was introduced as $V_{\mathrm{mock}}$ by Witzel and Zaremsky, who combined Thompson's group with cactus groups.

We prove a Bauer-Poulsen dichotomy theorem for simplices of invariant measures associated with permutation groups. More precisely, let $G$ be a transitive group of permutations of a countable set $\mathcal{S}$, and let $H$ be the stabilizer of a point of $\mathcal{S}$. Let $\overline{G}$ and $\overline{H}$ denote their closures in the topology of pointwise convergence. Assume the Polish group $\overline{H}$ has relative Property (T) in $\overline{G}$. Then the simplex $\mathcal{M}_\mathrm{inv}(2^\mathcal{S})$ of invariant probability measures for the induced action $G\curvearrowright 2^\mathcal{S}$ is Bauer if and only if $\overline{G}$ has Property (T), and is Poulsen otherwise. This addresses some examples and questions considered by Austin. We deduce this result from a more general model-theoretic statement of independent interest. To this end, we initiate the study of existentially closed models in affine logic.

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that, for lattices in connected semisimple real Lie groups with finite center and no compact factors, the scalar-expectation case of the corresponding classification is equivalent to ordinary ITAP.

A finitely generated group is profinitely rigid among a class of finitely generated groups if it can be distinguished among this class by its set of finite quotient groups. This paper proves that all lattices in $\mathrm{PSL}(2,\mathbb{C})$ are profinitely rigid among themselves. In addition, for any lattice $\Gamma\le \mathrm{PSL}(2,\mathbb{C})$, it is proven that $\mathrm{Out}(\widehat{\Gamma})\cong \mathrm{Out}(\Gamma)$, where $\widehat{\Gamma}$ denotes the profinite completion of $\Gamma$.

A group is an E-group if every element commutes with each of its endomorphic images. Caranti asked whether a finite E-group can have nilpotency class three. We prove that the $3$-group of order $3^{84}$ introduced by Abdollahi, Faghihi, and Mohammadi Hassanabadi, and later shown by Abdollahi, Faghihi, Linton, and O'Brien to have the corresponding automorphism property, is an E-group. Let $P$ denote this group and put $V=P/\Phi(P)\cong \mathbb{F}_3^9$. The nine power relations of $P$ determine a linear map $q:V\longrightarrow\Lambda^2 V$. We prove that $q$ has no nonzero proper subspace $U$ satisfying $q(U)\subseteq\Lambda^2 U$. Since the image induced by any endomorphism of $P$ on $V$ has precisely this closure property, every endomorphism acts on $V$ either invertibly or trivially. The invertible case is the known A-group case. In the trivial case the image first lies in $\Phi(P)=P'$, and the power relations then force it into $\Omega_1(P')=Z(P)$. Thus every element commutes with every endomorphic image. The tensor rigidity is reduced to an exact finite calculation on the $9841$ points of $\mathrm{PG}(8,3)$.

We prove an upper bound on the Dehn function of a group $G$ acting cellularly, cocompactly, and without inversions on a simply connected CW complex $X$ in terms of the Dehn functions of the vertex stabilizers, the Dehn function of $X$, and the distortion of the edge stabilizers, provided that $X$ is either a tree or the stabilizer of each $2$-cell has finite index in the stabilizer of every edge in its boundary. This provides a Dehn function analogue of Brown's Theorem for finiteness properties and an answer to a question of Zaremsky in these cases. We also prove analogues of our result for higher Dehn functions when $X$ is a tree.

A variety of universal algebras is called Cross if it is finitely based, finitely generated, and has only finitely many subvarieties. A monoid is aperiodic if all its subgroups are trivial. In the present article, we show that a variety of aperiodic monoids is Cross if and only if it excludes, as subvarieties, a certain list of 22 almost Cross varieties. Consequently, this list of 22 varieties exhausts all almost Cross varieties of aperiodic monoids.

Resolving Babai's minimal Cayley graph problem, we construct finite minimal Cayley graphs with arbitrarily large chromatic number.

Much work in knot theory has consisted of categorifying, and thereby strengthening, knot invariants. We take the opposite approach: decategorification, more commonly called combinatorics. We introduce a technique that relates the dual braid group generators, the Hecke images of pure braids, and factorization problems in reflection groups to knot invariants. We prove that the $(a,z{=}0)$-HOMFLYPT polynomial can be computed as a solution to such a problem. This technique was motivated by Coxeter–Catalan combinatorics. For example, we give a new proof of EL-shellability of the noncrossing partition lattice using the image of the full twist in the Hecke algebra; in a surprising sort of combinatorial reciprocity, its inverse computes the homotopy type. Similarly, noncrossing partitions naturally arise from our construction applied to positive powers of the full twist, while cluster complexes come from the same construction applied to negative powers. In crystallographic type, we exploit the conjugacy of all Coxeter elements to give the first reflection subword models for rational noncrossing Catalan objects. Our reciprocity gives two models: one generalizing noncrossing partitions, and one generalizing clusters. Applying the same method produces two (rational) noncrossing parking models.

This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let $\Gamma<G$ be such that $\{g\in G:g\Gamma g^{-1}\leq\Gamma\}$ generates $G$ as a group, and suppose that both $\Gamma$ and $G$ have property $(T)$. If $\Gamma$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ and the group double $G \ast_{\Gamma} G$ are nonsofic. These hypotheses hold for explicit pairs of elementary groups over polynomial and Laurent polynomial rings, in which both groups are residually finite and Kazhdan.

Let $G$ be a group. A bijection $f\colon G\to G$ is called inverse ambiguous if $f^{-1}(x)=f(x)^{-1}$ for every $x\in G$. We prove that every infinite group admits an inverse ambiguous function. An inverse ambiguous automorphism can exist only on an abelian group. We classify the finitely generated and divisible abelian groups admitting such automorphisms. If $A\cong\Z^r\oplus T$, where $T$ is finite, then $A$ admits an inverse ambiguous automorphism if and only if $r$ is even and $T$ admits one. Together with Toborg's finite classification, this gives an explicit classification of the finitely generated abelian groups admitting such automorphisms. For a divisible abelian group $A\cong\mathbb Q^{(\kappa_0)}\oplus \bigoplus_p C_{p^\infty}^{(\kappa_p)}$, such an automorphism exists if and only if none of $\kappa_0$, $\kappa_2$, or $\kappa_p$ for $p\equiv3\pmod4$ is a finite odd cardinal. We also prove that if every proper subgroup of an infinite locally finite group $G$ admits an inverse ambiguous automorphism, then $G$ admits one that leaves every subgroup invariant. Finally, we prove that a residually finite group whose finite quotients admit inverse ambiguous automorphisms is abelian. However, we prove the existence of infinite residually finite abelian groups whose finite quotients all admit such automorphisms, but the group itself does not.

We characterize four groups of Clifford circuits for Calderbank–Shor–Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice. We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group. For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.

Nielsen and Soelberg exhibited three torsion-free groups G_1, G_2, G_3 carrying 8-element sets without unique products, and asked whether any of the group rings R[G_i], R a domain, is a domain. We record that for every commutative domain R the answer is affirmative for all three groups: each G_i is virtually nilpotent, so the theorem of Kropholler, Linnell and Moody applies over every field, in every characteristic, and commutative coefficients reduce to the fraction field. Every ingredient is in the literature except the nilpotent structure of the relevant finite-index subgroup of G_2, which is supplied by the certified computational model of our companion paper (arXiv:2607.19687). The question therefore remains open exactly for noncommutative coefficient domains, where the unique-product mechanism – the only known ring-independent one – is precisely what these groups are constructed to lack. As a complement we report machine-checkable, DRAT-certified verifications that F_2[G_i] has no zero divisors with both supports in explicit balls of the defining generating sets, obtained by propositional reasoning alone, independent of the K-theoretic machinery; we state precisely what these certificates do and do not add.

We prove that a Kazhdan group admitting a sofic embedding into a metric ultraproduct of symmetric groups with a centralizer that acts ergodically on the associated Loeb probability space is locally embeddable in finite groups (LEF). In particular, every finitely presented Kazhdan group admitting such an embedding is residually finite. The main technical theorem says that the centralizer of a sofic embedding of a Kazhdan group is itself a metric ultraproduct of permutation groups.

Two characterizations of geometric finiteness in terms of weaker conditions are known for actions on Gromov hyperbolic spaces. In this paper, we establish analogous characterizations for general convergence group actions. To this end, we introduce the notion of an orbit uniform metric. We prove that a point is either conical or bounded parabolic if and only if its orbit is discrete with respect to an orbit uniform metric. As a consequence, we characterize geometric infiniteness in terms of the uncountability of the set of non-conical limit points. We further characterize geometric infiniteness by the existence of escaping sequences of hyperbolic elements, thereby extending the corresponding result for actions on Gromov hyperbolic spaces.

The classical Blaschke–Lebesgue theorem identifies the Reuleaux triangle as the planar convex body of constant width with minimum area. We investigate this extremal problem under prescribed symmetry constraints. Specifically, we classify the minimum-area convex bodies of constant width that are invariant under a finite group $G$ of isometries of $\mathbb{R}^2$ fixing the origin. For the exceptional reflection group $D_1$, the minimizers are precisely the Reuleaux triangles invariant under the prescribed reflection. If $G$ contains the half-turn $\mathcal R_\pi$, the disk is the unique minimizer. For odd $n\geq 3$, the minimizers are regular Reuleaux $n$-gons, unique up to rotation in the cyclic case $C_n$, and exactly those satisfying the prescribed reflection symmetry in the dihedral case $D_n$.

We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let $X=G/K$ be a symmetric space of noncompact type and real rank at least $2$. We prove a general vanishing theorem for the normalized torsion in first homology along sequences of torsion-free lattices in $G$. In particular, if $G$ is simple, we get \[ \dfrac{\log |H_1(M_n;\mathbb{Z})_{\operatorname{tors}}|}{\mathrm{vol}(M_n)} \longrightarrow 0 \] for any sequence of distinct manifolds $M_n = \Gamma_n \backslash X$. This answers a question of Abért, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. In addition, we get quantitative bounds with respect to the minimal injectivity radius for both the torsion in first homology and the minimal number of generators of $\Gamma$. Finally, we prove the analogous statements for affine buildings.

We classify, up to isomorphism, gradings by finite cyclic groups on classical simple Lie algebras $\mathfrak{g}$ over an algebraically closed field of arbitrary characteristic. Using the smoothness of the automorphism group scheme $\operatorname{\mathbf{Aut}}\mathfrak{g}$ and correspondence between $\mathbb Z_m$-gradings and morphisms $\boldsymbol{\mu}_m\to\operatorname{\mathbf{Aut}}\mathfrak{g}$, we express the classification as an orbit problem for certain Weyl-type groups. More generally, for the affine group scheme $\mathbf G$ associated to a semisimple algebraic group $G$ and the constant group scheme $\mathbf \Gamma_0$ associated to a subgroup $\Gamma_0$ of the automorphism group of the based root datum of $G$, we consider the classification of morphisms $\boldsymbol{\mu}_m\to \mathbf G \rtimes \mathbf\Gamma_0$ up to conjugation by $G \rtimes \Gamma_0$. We show that the classification in characteristic $p$ is the same as in characteristic $0$ except that, in characteristic $p$, only elements of $\Gamma_0$ whose order is prime to $p$ can occur. For $\mathbb{Z}_m$-gradings on $\mathfrak{g}$, this extends the classification by Kac coordinates to arbitrary characteristic, with the caveat that only diagram automorphisms of order prime to $p$ are allowed and, if $p=2$ or $3$, the type of $\operatorname{Aut}\mathfrak{g}$ is not always the same as the type of $\mathfrak{g}$.

Let $G$ be a connected real semisimple linear algebraic group with finite center and no compact factors, let $X$ denote its associated Riemannian symmetric space, and let $h_{\mathrm{vol}}(X)$ be the volume growth entropy of $X$. We show that there exists an $\epsilon = \epsilon(G) > 0$ so that the following holds: if $\Gamma < G$ is a discrete subgroup with critical exponent greater than $h_{\mathrm{vol}}(X) - \epsilon$, then $\Gamma$ is Zariski dense in $G$. If $G$ is further assumed to be isomorphic to one of the isometry groups of the real, complex, or quaternionic hyperbolic spaces, we determine the smallest possible value of the critical exponent to guarantee Zariski density of the associated subgroup (in other words, the largest possible value of $\epsilon = \epsilon(G)$). In the setting of discrete subgroups of real semisimple Lie groups with no compact factors, this generalizes Borel's Density Theorem for lattices. The key ingredient is input from the theory of unitary representations, particularly the recent work of Benoist–Liang (which was inspired by earlier work of Benoist–Kobayashi) on general temperedness criteria for the quasi-regular representation of $L^2(G/H)$ for any closed subgroup $H$ of $G$.

This paper adapts Rump's correspondence between radical rings and braces to a more general setting, utilizing a suitable class of near-rings. Given a right near-ring $(R,+,\circ)$ with a multiplicative identity, we define a new operation that yields a monoid. Under natural compatibility conditions expressed via a filtration, this monoid becomes a topological group, yielding a topological right skew brace. This generalization recovers radical-ring braces and successfully applies to near-rings of maps under composition. We provide several explicit applications, demonstrating that the Nottingham group, groups of triangular functions, iterated wreath products of arbitrary groups, and groups of IA-automorphisms of free nilpotent groups naturally arise as multiplicative groups of such topological skew braces.

We present a new proof of the theorem of Dickson that gives generators for the groups $SL_{2}(q)$ where $q$ is an odd prime power.