Papers

We reduce the Alperin-McKay-Navarro conjecture to an inductive condition on the universal covering groups of non-abelian finite simple groups. The reduction yields height-zero character bijections compatible with Brauer correspondence and block relations between H-triples for arbitrary finite ambient groups. We record the resulting arithmetic and structural consequences, including preservation of $p$-rationality levels and character-theoretic criteria for Sylow subgroups. The proof combines semilinear centralization and transfer through quasisimple components with induction on the central index. It uses the Clifford and gluing theorems for block relations and a central-defect Dade-Glauberman-Nagao correspondence established in two related papers. We also verify the inductive condition for several sporadic groups at primes for which the Sylow subgroups have prime order, and in defining characteristic for the Suzuki groups ${}^{2}B_{2}(2^{2m+1})$ with $m \geq 2$ and the small Ree groups ${}^{2}G_{2}(3^{2m+1})$ with $m \geq 1$.

We give a direct, self-contained construction of the three sporadic Fischer groups $\mathrm{Fi}_{24}'$, $\mathrm{Fi}_{23}$, and $\mathrm{Fi}_{22}$ from the $783$-dimensional Conway-Parker algebra. We prove that its distinguished roots define involutory algebra automorphisms whose projective actions generate the full Fischer $3$-transposition group $\mathrm{Fi}_{24}$. Its commutator subgroup gives $\mathrm{Fi}_{24}'$, while $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ arise as centralizer quotients associated with one and two commuting transpositions. The root and frame geometry determines the group orders and leads to elementary proofs of simplicity, as well as natural rank-three actions and nonsplit central extensions. The construction uses standard facts about the Golay code, Parker's loop, and $M_{24}$. It does not use the Monster or previously known existence or order results for the Fischer groups. Fischer's classification and later recognition theorems are used only for the final identification.

We construct maximal subgroups of topological full groups and their commutator subgroups using factor maps and wide open subgroupoids of minimal effective ample groupoids with Cantor unit spaces, assuming almost finiteness or pure infiniteness. For two-to-one coverings and prime factor maps with principal target and fibers of cardinality at most two, surjectivity and injectivity of the same induced map on full-group abelianizations characterize maximality of the factor full group and factor commutator subgroup, respectively. For subgroupoids arising from prime-order cyclic actions free on the unit space, the normalizer of the smaller commutator subgroup in the ambient full group is a semidirect product, maximal exactly when the inclusion induces a surjection on full-group abelianizations. In both constructions, the embedded commutator subgroup has a unique maximal overgroup in the ambient commutator subgroup, namely its normalizer. For Cantor minimal systems, the factor construction yields partition stabilizers of the type in the Grigorchuk-Vorobets conjecture. The subgroupoid normalizers act topologically primitively. Groupoid homology makes the criteria computable for these systems and shifts of finite type. Examples include finitely generated nonsimple and locally finite simple maximal subgroups of infinite index in finitely generated infinite simple amenable groups.

We study mixed symplectic sums along connected unbranched cyclic multisections of degrees $p,q\geq 2$ in ruled surfaces. With \[ p(g-1)=q(h-1)=n, \] the resulting four-manifold has signature zero and Euler characteristic \[ e=4n\left(1-\frac1p-\frac1q\right). \] We classify the types having the rational cohomology ring of $\#_k(S^2\times S^2)$. They occur only for odd $k$ and, with $t=(k+1)/2$, $a=g-1$, and $b=h-1$, are exactly the solutions of \[ n=a+b+t,\qquad a\mid b+t,\qquad b\mid a+t. \] For each odd $k$ there are finitely many types; we give an explicit parametrization and tables for $k\leq 15$. Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is $kH$. We prove minimality, boundary incompressibility, an integral Mayer–Vietoris presentation, and a regular cyclic cover of degree $\operatorname{lcm}(p,q)$. The boundary subgroup $\pi_1(\Sigma_G)\times\mathbb Z$ embeds in the fundamental group; this gives a nontrivial Bass–Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for $k=3$ and seven for $k=5$; for $k=1$ we recover the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$.

Given a finite transitive group $G\leq \operatorname{Sym}(\Omega)$, the derangement graph $\Gamma_G$ is the graph whose vertex set is $G$, and two vertices $g$ and $h$ are adjacent if the ratio $h^{-1}g$ is a fixed-point-free permutation. In this paper, we show that the automorphism group of the derangement graph of the transitive permutation group corresponding to the natural action of $\operatorname{PGL}_{2}(q)$ on the projective line $\operatorname{PG}_{1}(q)$ is \begin{align*} \operatorname{Aut}(\Gamma_{\operatorname{PGL}_{2}({q})}) = \left(L_{\operatorname{PGL}_2(q)}\times R_{\operatorname{PGL}_2(q)}\right) \rtimes \left(\langle \psi \rangle \times \gamma_{\operatorname{Aut}(\mathbb{F}_q)}\right), \end{align*} where $L_{\operatorname{PGL}_2(q)}$ is the left-regular representation of $\operatorname{PGL}_2(q)$, $R_{\operatorname{PGL}_2(q)}$ is the right-regular representation of $\operatorname{PGL}_2(q)$, $\gamma_{\operatorname{Aut}(\mathbb{F}_q)}$ is the group of conjugation by elements of $\operatorname{Aut}(\mathbb{F}_q)$, and $\psi: \operatorname{PGL}_2(q) \to \operatorname{PGL}_2(q)$ such that $\psi(x) = x^{-1}$.

This mostly expository paper gives a unified treatment of many interpolation results for bandwidth-limited functions on locally compact abelian groups, based on reproducing kernel Hilbert spaces. It includes an abstract interpolation theorem phrased in terms of frames.

We prove that any block of a profinite group with a topologically finitely generated defect group $D$ is isomorphic to a block of a virtually pro-$p$ group, and if $D$ is finitely presented then so is the basic algebra of the block over $k$. Using these general results we then prove that any block with defect group $D=\mathbb Z_p^n$ is Morita equivalent to $\mathcal O_\alpha[[\mathbb Z_p^n\rtimes E]]$ for a finite $p'$-group $E$ acting faithfully and a $2$-cocycle $\alpha\in H^2(E,k^\times)$. In particular, Donovan's conjecture holds for such blocks of profinite groups.

In this paper, we prove that for $\ell>5$, the number of $\ell$-regular conjugacy classes of a finite group is at least that of the normalizer of a Sylow $\ell$-subgroup.

We study matrix semirings over the three-element flat additively idempotent semiring with elements one, a, and infinity, where the square of a is infinity. We determine the associated matrix-variety chain completely. The scalar, two-by-two, and three-by-three cases generate three distinct varieties, while every matrix dimension at least three generates the same variety. The proof shows that every failure of an identity in arbitrary dimension is already witnessed on three indices, and it yields a coordinate criterion for all identities in the stable variety. We also realize every graph semiring arising from a directed graph of in-degree and out-degree at most one as a divisor of a direct power of the two-by-two matrix semiring. Consequently, the variety generated by all three-nilpotent flat semirings is contained in the two-by-two matrix variety. Directed cycles and independent reversal identities then embed the power-set lattice of the odd primes into each interval between the base variety and a nontrivial matrix variety. Thus every such interval has continuum cardinality and contains continuum-sized chains and antichains. Finally, we determine the last three powers of the multiplicative subsemiring obtained by deleting the constant all-one matrix, and we develop general matrix operators on the lattice of additively idempotent semiring varieties, including stable closures, stable cores, and propagation of equality along matrix-dimension chains.

An explicit finite identity basis is given for the eight-element semiring of upper triangular Boolean \(2\times2\) matrices in the signature \((+,\cdot)\). The basis consists of a known multiplicative basis, the ai-semiring laws, and 30 mixed identities, each using at most eight variables. The proof combines a finite basis for the multiplicative reduct with finite rules for shuffling words, duplicating a marked occurrence, and interchanging adjacent occurrences while adding prescribed witnesses. This converts the one-letter gap criterion into a derivation of every valid semiring identity. We also study the band subvariety, a distinguished 156-element interval of the subvariety lattice, congruences, flat members, and finite representations of free algebras. Every \(m\)-letter word has an equivalent subword of length at most \(m^2\) for \(m\geq2\), and the free algebras have doubly exponential rank growth. For arbitrary partially ordered sets, we determine the equational theory of Boolean relation semirings and of finite-support incidence semirings over nontrivial bounded distributive lattices: finite height \(h\) gives the theory of \(T_h\), while unbounded height gives precisely the identities of all additively idempotent semirings.

This paper is the fourth in a series devoted to the finite basis problem for $4$-element additively idempotent semirings. These algebras are divided into five types according to their additive reducts; we study the largest of these types, namely those whose additive reducts are chains. Up to isomorphism, there are $386$ such algebras, denoted by $S_{(4, k)}$, $481 \leq k \leq 866$. We show that all but three of them, namely $S_{(4,545)}$, $S_{(4,634)}$, and $S_{(4,710)}$, are finitely based. This completes the classification, with respect to the finite basis property, of all $4$-element additively idempotent semirings with chain additive reducts.

An $\mathfrak{N}$-semigroup is an archimedean, idempotent-free, cancellative and commutative semigroup. For example, the additive semigroup $\mathbb{R}_{>0}$ of strictly positive real numbers is a $\mathfrak{N}$-semigroup. Hewitt-Zuckerman showed that every $\mathfrak{N}$-semigroup admits a semigroup homomorphism to $\mathbb{R}_{>0}$. We show that every uniquely divisible $\mathfrak{N}$-semigroup $S$ admits enough semigroup homomorphisms to $\mathbb{R}_{>0}$ to detect divisibility in $S$. %They were studied by Tamura and Kobayashi. The main ingredient is a version of the Eidelheit and Kakutani hyperplane separation theorem for vector spaces over $\mathbb{Q}$. A counterexample due to Ravsky explains why this result does not generalise to the larger class of $\mathfrak{N}$-semigroups.

We present a constructive atlas of finite simple groups with proofs of their orders, simplicity, and structural properties. The eight completed families are cyclic groups of prime order, alternating groups, the classical series $A_r(q)$, $B_r(q)$, $C_r(q)$, $D_r(q)$, and the exceptional series $G_2(q)$ and the small Ree groups $ {}^2G_2(3^{2m+1})$, $m\geq1$. The fifteen sporadic entries are $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$, $M_{24}$, $\mathrm{Co}_1$, $\mathrm{Co}_2$, $\mathrm{Co}_3$, $\mathrm{McL}$, $\mathrm{HS}$, $\mathrm{Suz}$, $J_2$, and $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$, $\mathrm{Fi}_{24}'$. The simple parameter ranges and exceptional cases are stated explicitly. The models arise from codes, lattices, forms, algebras, and finite geometries, including the split octonions and the Conway–Parker algebra. They retain natural actions, stabilizers, central quotients, and comparison maps for subsequent group theory. Several classical comparison isomorphisms relate the models, while involution-class counts distinguish the equal-order orthogonal and symplectic families in odd characteristic and rank at least three. Drawing on classical sources and companion mathematical work, the project develops the construction side of finite simple group theory, not the exhaustiveness proof of the classification. The completed models and stated proofs are formalized in Lean for verification and reuse.

We show that there are infinitely many k for which Sp_{2k}(2) is not generated invariably by two elements of prime order.

We study the profinite non-rigidity of arithmetic lattices and its implications for the Kähler property. In the first part, we characterize the absolute Dynkin types that admit non-isomorphic real forms of higher-rank Lie groups containing torsion-free arithmetic lattices with isomorphic profinite completions. In the second part, as a geometric application, we answer a question asked independently by Arapura and Libgober, by showing that the Kählerness of finitely presented, residually finite groups is not determined by its profinite completion.

We show that there exists a constant $\epsilon > 0$ such that if $C$ is a conjugacy class in $A_n$ of size at least $|A_n|^{1-\epsilon}$, then $A_n \setminus \{1\} \subseteq C^2$. This proves the robust version of Thompson's conjecture for the alternating group, conjectured by Shalev (Annals of Math., 2009). Our proof combines character bounds and combinatorial cancellation techniques with the recent theory of hypercontractivity for functions over symmetric groups.

This paper studies Veech groups of dilation surfaces whose fundamental group is not finitely generated. We prove that if $ S$ is a surface with self-similar end space and every end is accumulated by genus, then every countable subgroup of $SL(2,\mathbb{R}) $ can be realized as the Veech group of a dilation surface homeomorphic to $ S$. Additionally, we can construct this dilation surface such that it realizes horizon saddle connections in a prescribed set of directions. This contrasts with the case of closed dilation surfaces, where horizon saddle connections impose restrictions on the algebraic structure of the Veech group.

A group is called an $n$-$\bfc$-group if each of its conjugacy classes is finite and contains at most $n$ elements. By a theorem of B.~H.~Neumann the derived subgroup of such a group is finite of $n$-bounded order, whereas the order of the group itself need not be bounded, as extraspecial $p$-groups show. We prove that an $n$-$\bfc$-group with trivial centre is finite of order at most $n^{64(\log n)^{5}}$. Passing to the quotient by the hypercentre, we deduce that no assumption on the structure of the group is needed at all: for an arbitrary finite $n$-$\bfc$-group $G$ one has $|G:Z_\infty(G)|<n^{64(\log n)^{5}}$. The finiteness of the group is required here only for the correct definition of the hypercentre.

We prove that nilpotent and metabelian products of groups with decidable word problem have decidable word problem. In contrast, we exhibit families of groups with decidable word problem for which their solvable product have undecidable word problem. In doing so, we exhibit hyperbolic groups with undecidable membership problem in its derived subgroups.

Let $\Gamma$ be a finitely generated hyperbolic group, and let $Q$ denote the conformal dimension of its Gromov boundary $\partial\Gamma$. We prove that for every $p < \frac{Q}{Q-1}$, the group $\Gamma$ admits a proper uniformly Lipschitz affine action on $\ell^p$ with compression exponent $1/p$. The $\ell^p$-space hosting the action is designed from a Markov chain on a specialized hyperbolic filling of $\partial\Gamma$.

We introduce a notion of topological complexity of a group with respect to a family of subgroups, with the case of the trivial and diagonal families recovering the classical category and topological complexity, respectively. These invariants come equipped with a battery of upper and lower bounds derived from functoriality and Bredon cohomology. We introduce a new family of subgroups, the permutational family, and show that the corresponding invariant is a lower bound for the distributional topological complexity of Dranishnikov-Jauhari and Knudsen-Weinberger. As a first application, we show that the permutational and diagonal families coincide, and thus that classical and distributional topological complexity are equal, for a large class of torsion-free groups, extending work of Dranishnikov. Second, we show that the lower bounds of Grant-Lupton-Oprea are in fact lower bounds on distributional topological complexity; in particular, it follows that Farber's conjecture holds distributionally.

We prove that almost sure topological finite generation holds for a general class of random models of profinite groups. We deduce that in the models introduced by Liu–Wood and Sawin–Wood, one has that finite presentation holds almost surely, with almost sure control of the deficiency in the presentation. In particular this settles questions raised in work of Liu–Wood and Sawin–Wood. Using the same underlying principle, we show that there exists a unique universal d-generated perfect profinite group: its finite quotients are precisely the finite d-generated perfect groups. This settles a conjecture of Nikolov. We show in addition that this group is projective and admits a profinite presentation with $d$ generators and $d$ relations, and with no fewer relations on $d$ generators. The main underlying theme is to bring in an insight from crown theory: groups with high rank are always witnessed by a crown with large multiplicity. This approach was discovered independently by ChatGPT5.5 pro and Aletheia, an internal agent at Google DeepMind.

This note record a finitely generated extension of Caprace's example of a non-inner-amenable group is also not properly proximal, which answers the original question by Boutonnet, Ioana and Peterson. The extension is an explicit ICC subgroup of $\operatorname{GL}_4(\mathbb{F}_2(t))$, which has no property (T). Failure of proper proximality follows from the work of Ishan-Peterson-Ruth and the passage of proper proximality to co-amenable subgroups. Non-inner-amenability follows from a property-(T) subgroup and an elementary centralizer computation.

In this article, we prove that the theory $T_3$ of groups of exponent $3$ has a model companion. Previous work established the existence of model companions for theories of groups of fixed finite exponent and nilpotency class at most $2$ (Saracino–Wood), and for theories of groups of prime exponent $p$ and nilpotency class at most $c<p$ (Maier). These results do not cover $T_3$, since groups of exponent $3$ may have nilpotency class $3$. Our theorem verifies the $n=3$ case of a conjecture proposed by the third author that, for each integer $n>1$, the theory $T_n$ of groups of exponent $n$ has a model companion if and only if every finitely generated group of exponent $n$ is finite, or equivalently, if and only if the Burnside problem has a positive solution for exponent $n$.

Given two (closed) subgroups H, K of a profinite group G, we write Pr(H,K) for the probability that a random pair from HxK commutes. Here we are concerned with profinite groups containing Sylow subgroups P, Q such that Pr(P,Q) > 0. First, we show that if G is a profinite group containing a Sylow 2-subgroup L and a Sylow p-subgroup P , where p is odd, such that Pr(L,P) is positive, then G is virtually pro-p-soluble (Theorem 1.1). Then we handle similar issues for profinite groups admitting coprime automorphisms. In particular, we prove that if G is a profinite group admitting a group of coprime automorphisms A such that there is a Sylow 2-subgroup L of C_G(A) and an A-invariant Sylow p-subgroup P of G, where p is odd, for which Pr(L,P) > 0, then G is virtually pro-p-soluble (Theorem 1.3). On the other hand, if we only have Pr([L,A],[P,A]) > 0, then [G,A] need not be virtually pro-p-soluble. We show that in this case [G,A] has an open normal subgroup of non-p-soluble length at most 1 (Theorem 1.4). Furthermore, if P is an A-invariant Sylow p-subgroup of G such that Pr([P, A],[P, A]^x) > 0 for every x in G, then [G,A] has an open normal subgroup of non-p-soluble length at most 1 (Theorem 1.5).

Asymptotic cohomology was recently introduced to study stability problems in group theory. This paper studies vanishing and Hausdorffness questions around asymptotic cohomology. Our first main result is that, in a fixed degree and for a fixed asymptotic coefficient module, vanishing of asymptotic cohomology is equivalent to vanishing of reduced asymptotic cohomology. In the special case of diagonal coefficient modules, we also obtain a full characterisation of vanishing in terms of bounded cohomology. Our second main result is that asymptotic cohomology vanishes in degree 1 for uniformly convex asymptotic Banach modules. From this, we can deduce Hausdorffness of degree-2 bounded cohomology under the same uniform convexity assumption.

We prove that $\mathrm{C}^*$-simple groups which admit a faithful extreme boundary are $\mathrm{C}^*$-selfless. More precisely, we establish complete selflessness of the canonical inclusion of the reduced group $\mathrm{C}^*$-algebra into the reduced crossed product. In particular, the Higman–Thompson groups and the Brin–Thompson groups are $\mathrm{C}^*$-selfless.

We study units in group rings of the Hantzsche-Wendt group. We construct a compatible system of non-trivial unitary units over $\mathbb Z/2^n\mathbb Z$ for every $n\ge1$, and “almost-group” units over $\mathbb Z/2\mathbb Z$, meaning non-trivial units $u$ with the property that $u-g=u^{-1}-g^{-1}$ for some group element $g$.

We prove that a lattice $\Gamma$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb C)$ has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attributed to Sarnak by Geninska and Leuzinger in 2008. The same characterization by quaternion algebras holds if the difference set of traces is not dense. Our method also gives a similar result for lattices in $\mathrm{SL}_d(\mathbb R)$, for every $d\ge3$: a lattice $\Gamma<\mathrm{SL}_d(\mathbb R)$ has positive trace gap if and only if all its traces are integers. Equivalently, after conjugation, it has finite index in the norm-one group of an order in a central simple algebra of degree $d$ over $\mathbb Q$ that splits over $\mathbb R$. We also discuss spectral consequences and other applications of the results and techniques.

We prove that the language of palindromic reduced words in a Coxeter system is regular if and only if the Coxeter group is a direct product of a finite Coxeter group with finitely many copies of the infinite dihedral group.