Papers

We classify, up to quasi-isometry, the large-scale geometry of Galton–Watson trees for every finitely supported offspring distribution. Apart from the trivial finite diameter regimes, we condition on infinite diameter. We find that the remaining classes are the ray class, the full tree class (which includes the binary tree), one chain class $(\mathrm{C}_\Lambda)$ for every possible branching semigroup $\Lambda$, and the bushy class. Two independent trees almost surely admit a root-preserving quasi-isometry when their offspring distributions belong to the same class and are almost surely not quasi-isometric when they belong to different classes. Any two survival-conditioned supercritical realisations with finitely supported offspring laws nevertheless a.s.~admit quasi-isometric embeddings in both directions. We prove that the class of a realisation depends only on the support of the offspring distribution, not its specific distribution. For offspring distributions supported on $\{1,2\}$, we additionally obtain an explicit exponential tail bound for the probability of non-existence of a root-preserving $D$-quasi-isometry. The classification also implies that the corresponding random Cantor boundaries are almost surely quasisymmetrically equivalent. Conditioned on nonextinction, this applies to strongly separated fractal percolation, even when the underlying self-similar iterated function systems and retention parameters differ. We also classify two families of trees with continuous random branching times. Our proofs use new automorphism-matching theorems for random graph labellings of trees, based on contraction estimates for mismatch potentials. These matching results for Markov labellings on the binary tree may be of independent interest. The quasi-isometry classification, embeddability, and matching theorems are formally verified in Lean.

We study homological Bogomolov multipliers and finitary Bogomolov multipliers of infinite groups, with emphasis on polycyclic groups. We obtain structural descriptions for torsion-free nilpotent groups of class two, semidirect products, wreath products, and infinite simple groups. The class-two case reveals a connection with rational points on linear sections of Grassmannians, explaining an arithmetic obstruction to a general algorithm. On the positive side, we give a deterministic algorithm for virtually abelian polycyclic groups and an algorithm for finitary Bogomolov multipliers of arbitrary polycyclic groups, both implemented in GAP.

We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation $\Theta(n^0)$ short closed geodesics, potentially $\Theta(n^0)$ pairs of nearby order-2 cone points, in general $\Theta(n^{\frac{1}{m}})$ cone points of order m, and with high probability no small regions with more complicated topology. This generalises many results of Magee and Puder (2023) from orientable surfaces to possibly non-orientable orbifolds, and answers some of the questions raised by Puder and Zimhoni (2024).

We provide direct constructions of nontrivial root elations of thick generalized quadrangles embedded in thick polar spaces of higher finite rank and show that all of these can be expressed as self-projectivities of length $4$. This reveals a connection between the little projective groups and the special projectivity groups of thick polar spaces of finite rank explicitly. We also prove that the constructed root elations extend to the ambient polar space. Furthermore, we introduce point elations and line elations and use them to give an equivalent definition of the Moufang property for thick polar spaces, without using the notion of apartments. This, in turn, allows us to define the Moufang property for thick polar spaces of infinite rank. Consequently, we show that all thick polar spaces of finite and infinite rank are Moufang.

For n>=2, we prove a rigidity theorem for the Hermitian boundary kernels L_{t,s}(x,y)=|1-<x,y>|^t exp(is arg(1-<x,y>)), x != y, on S^{2n-1}, with L_{t,s}(x,x)=0, t>0, and s in [-1,1]. Every finite Gram matrix of L_{t,s} has positive index at most one if and only if 0<t<=1 and s=+-t. The proof combines Fourier analysis on boundary circles with restrictions to two orthogonal complex directions. The dimension threshold is sharp. By normalizing the Gram kernels of equivariant boundary maps, we show that the translation-length factor t and the signed Cartan factor s of every continuous irreducible representation from PU(n,1) to the holomorphic isometry group of infinite-dimensional complex hyperbolic space satisfy s=+-t. Combining this with the known endpoint rigidity, Monod's constructions, and Ruiz Stolowicz's complete-invariant theorem yields the classification: the holomorphic conjugacy classes are parametrized by (t,epsilon) in (0,1) x {+-1}. Allowing anti-holomorphic conjugacy identifies the two signs. Using Monod's automatic-continuity argument and the same boundary rigidity, we also classify all irreducible self-representations of the full isometry group of infinite-dimensional complex hyperbolic space. Up to conjugacy, these are precisely Monod's representations with parameter 0<t<=1. In particular, the canonical-orbit hypothesis in Monod's classification theorem is automatic.

Let $\Avar_n$ and $\VV_n$ be the varieties generated by all $n$-element additively idempotent semirings (ai-semirings) and all $n$-element semirings with commutative addition, respectively, in the signature $(+,\cdot)$ without constants. We prove that both varieties are finitely based if and only if $n\leq2$. The five-element case follows from cyclic identities and a structural analysis of products of squares. For $n\geq6$, each variety contains a finite inherently nonfinitely based ai-semiring. We also prove $\VV_n\cap\AI=\Avar_n$ for every $n$, where $\AI$ is the variety of all ai-semirings.

We reconstruct the generalized hexagon of order $(2,8)$ from an abstract even rank-$26$ lattice of determinant $3$ and minimum $4$, and conversely reconstruct the lattice from the hexagon. Harmonic theta identities determine the $819$ shortest vectors in a non-zero discriminant class and their association scheme, and show that every non-empty positive-norm shell of the lattice and its dual is a spherical $5$-design. In the converse direction, the rank-$26$ idempotent gives the lattice, with saturation proved by a short dual-coset argument. A positive-definite Niemeier construction proves existence and uniqueness of the lattice and hence of the hexagon. The third moment of a projective $3$-design in the Cayley plane reconstructs the traceless Albert product. This upgrades angle-preserving bijections to elements of $F_4(\mathbb R)$ and proves geometric uniqueness of the tight $819$-point projective $5$-design. Finally, line deletion gives a rootless index-four sublattice of $N(A_1^{24})$ with a Golay trio and an oriented gluing. The gluing data form eight root-sign orbits, on which $L_3(2)\cong L_2(7)$ acts as on $\mathbf P^1(\mathbf{F}_7)$; two explicit Golay permutations prove transitivity. This proves uniqueness and gives the automorphism-group order. The group is identified afterward as ${}^3D_4(2):3$, with a central factor $C_2$ for the full lattice group. The reconstruction and deletion arguments parallel the length-$26$ binary code construction.

We extend the explicit octonionic approach to groups of type $\mathrm{E}_6$ to the twisted groups ${}^2\mathrm{SE}_{6,K}(F)$ associated with a quadratic Galois extension $K/F$. These groups act on the Albert space over $K$, preserving the Dickson–Freudenthal determinant and a Hermitean form. When the field norm $K^{\times}\to F^{\times}$ is surjective, we give explicit generators and coördinate proofs of the three-orbit theorem on white points and of the stabilisers of representative white vectors. Under this hypothesis we also prove simplicity of the central quotient. Over finite fields we recover the classical orbit lengths and group orders from the stabiliser orders and the total number of white points. Finally, without assuming norm surjectivity, we show that the two isotropic white-point orbits remain unchanged, while the non-isotropic orbits are parametrised by $F^{\times}/\mathrm{N}_{K/F}(K^{\times})$.

For braces, associativity of the $\star$-product is equivalent to two-sidedness. We extend this result to skew braces by proving that any two of the following conditions imply the third: two-sidedness, associativity of the $\star$-product, and centrality of $A\star A$ in the additive group. We then study two-sided skew braces with associative $\star$-product through the radical ring associated with their additive abelianization. This yields a canonical fibre-product description modulo an annihilator ideal, exact sequences relating higher $\star$-products to the powers of the radical ring, coincidence of the left, right and strong $\star$-series, and sharp bounds on their nilpotency indices. We also obtain an explicit description of the lower central series, classify the structures whose additive abelianization is infinite cyclic, and characterize when they are bi-skew.

Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. For every $r\ge1$, we prove that a product-one-free sequence of length $2r+11$ over $E_2\times C_3^r$ cannot contain exactly $2r-1$ central terms. Thus the critical layer with twelve noncentral terms is excluded. The proof combines a relative moment criterion for abelian normal subgroups with a finite theorem in a symplectic four-space over $\mathbb F_3$. Under explicit subspace occupancy bounds, the family of balanced triangles that can actually be completed to a nonfull zero-sum block admits edge weights summing to one on every triangle. A dual cycle argument reduces this assertion to a potential condition on branching edges. All remaining configurations contain one of eleven minimal frames; two separately implemented exhaustive checks verify all their admissible extensions. The unrestricted check has 9544046 leaves. Complete source code and execution records are supplied. The exact value of $\mathsf{d}(E_2\times C_3^r)$ for arbitrary $r$ is not determined by this result.

We give a unified arithmetic and geometric treatment of the Hall-Janko simple group and the icosian Leech lattice. Let $\mathcal{I}$ be the standard icosian maximal order over $K=\mathbb{Q}(\sqrt5)$. Following the mass-formula and Venkov framework for the Niemeier and Leech lattices, we classify positive definite unimodular Hermitian $\mathcal{I}$-lattices of quaternionic rank $3$. Hashimoto's mass formula, the Siegel-Weil average, and Gundlach's structure theorem determine the scalar theta series of every nonsplit class. A two-place harmonic separation of the Leech minimal shell proves that its three Hermitian components are quaternionic projective $5$-designs. For the norm-$2$ shell, integrality and projective moments recover the complete angle scheme and its $525$ orthogonal frames. The finite code of one frame then saturates the residual mass, giving exactly the split and Tits classes and an independent computation of the order of $\operatorname{Aut}_{\mathcal{I}}(L_{\mathrm T})\cong2.J_2$. Finally, a quaternionic line system with the prescribed five projective inner products has at most $315$ points, with equality precisely for the Hall-Janko configuration up to $P\operatorname{Sp}(3)$. Equality in Hoggar's bound supplies the design property; Cohen-Tits uniqueness, golden-angle fission, and the centered cubic moment give geometric rigidity.

We give a negative answer to Problem 17.39 of the Kourovka Notebook. For every odd prime power $q$ and every $m\geq1$, we construct a finite metabelian group $G_{m,q}$ with $Z_\infty(G_{m,q})=1$ and $|\Phi(G_{m,q})|=q$ for which the least number of system normalizers with trivial intersection is $m+1$. Thus the number is unbounded even when the order of the Frattini subgroup is fixed.

We construct explicit examples of skew braces $A$ for which the third term $A^3$ in the left series is not generated by the right-normed star products $a_1*(a_2*a_3)$ of length $3$ as a sub-skew brace.

Let $M$ be the six-element commutative semigroup occurring as the common multiplicative reduct of the semirings $SR_6$ and $TR_6$. The closing paragraph of Shao, Ren, and Gao~\cite{ShaoRenGao2026} asks for the finite-basis and subvariety questions for the four remaining compatible additions on $M$. We answer these questions for the four isomorphism types $R_{01},R_{02},R_{11},R_{12}$. First, we classify all compatible additions on $M$: there are nine labelled additions and six isomorphism types, parametrized by $R_{ij}$ with $0\leq i\leq j\leq 2$. For each of the four new types we give a graph-theoretic criterion for every identity, an explicit infinite basis, and a proof of nonfinite basability. The generated varieties $\V(R_{01})$ and $\V(R_{02})$ have eleven subvarieties each, while $\V(R_{11})$ has sixty-six. The lattice $\Sub(\V(R_{12}))$ is countably infinite. Every identity in this variety reduces to a subset of twenty-five fixed identities together with two monotone path families $\gamma_n$ and $\gammaD_n$. This yields a canonical signature $(H,p,q)$, complete normal forms, explicit meet and join operations, and a formula for all covers. There are 153 fixed nodes, 43 one-parameter families, and 9 two-parameter families; exactly eighteen subvarieties are finitely based, and the unique limit subvariety is $\V(SR_6)$. The strong nonfinite-basis status of the four finite semirings remains open.

We study torus-exterior replacement in four-manifolds, with the construction controlled by the peripheral homomorphism. Using torus exteriors arising from the work of Boyle and Kanenobu-Kazama, with explicitly controlled fundamental groups and peripheral systems, we construct smooth homotopy 4-spheres and manifolds homeomorphic to S^2 x S^2 and \#_3(S^2 x S^2). We also study direct gluings of two torus exteriors. A swap gluing of two one-null exteriors kills both meridians and gives a simply connected manifold with Euler characteristic \(4\) and signature \(0\), while a meridian-preserving gluing of doubly-null exteriors retains the complement groups as an amalgam over the meridian subgroup. Twisted peripheral gluings give cyclic and explicit finite nonabelian fundamental groups. We also consider the full Kanenobu-Kazama \(1\)-handle family and Litherland tori with full-rank peripheral subgroup; the latter have no null primitive boundary slope and lead naturally to amalgamated-product fundamental groups.

We find all ring of integers, R, and finite groups G, such that RG has almost trivial units. Moreover, for such a group ring we determine the Swan subgroup of its locally free classgroup

The classical Thompson's conjecture asserts that in any finite nonabelian simple group $G$, there exists a conjugacy class $S$ whose square is equal to the entire group. Shalev (Annals Math., 2009) conjectured that a much stronger quantitative version holds: there exists $\epsilon > 0$ such that for any conjugacy class $S$ with $|S| \geq |G|^{1-\epsilon}$, we have $G \setminus \{1\} \subseteq S^2$. While the conjecture turns out to fail in general, we prove a close variant of the conjecture over $\operatorname{PSL}_n(2)$: There exist $\epsilon>0$ and $N \in \mathbb{N}$ such that for all $n\geq N$, every conjugacy class $S\subseteq\operatorname{PSL}_n(2)$ with $|S|\geq|\operatorname{PSL}_n(2)|^{1-\epsilon}$ satisfies $\{x\in\operatorname{PSL}_n(2):\operatorname{supp}(x)\neq1\}\subseteq SS^{-1}.$ On the other hand, we show that for each $\epsilon>0$, there exist conjugacy classes $S \subset \operatorname{PSL}_n(8)$ of size $>|\operatorname{PSL}_n(8))|^{1-\epsilon}$ for which $SS^{-1}$ misses even elements with support size linear in $n$. In addition, for every odd $n \ge 5$, we construct an inverse-closed normal set in $\operatorname{PSL}_n(5)$ with fixed positive density whose square misses a nonidentity element. This answers on the negative a question of Shalev (ECM, 2021) and disproves a conjecture of Pyber (Kourovka notebook, 2022).

We construct finite covers of a fixed product of two closed hyperbolic surfaces of genus two whose fundamental groups are generated by at most fifteen elements and whose injectivity radii tend to infinity. The construction uses fibre products over finite groups. These covers are closed aspherical four-manifolds with universal cover $\mathbb H^2\times\mathbb H^2$. They give counterexamples to a conjecture of Avramidi and Delzant for symmetric spaces of higher rank, in the case \(\mathbb H^2\times\mathbb H^2\).

We prove that right-angled Artin groups (RAAGs) are profinitely rigid relative to the class of Artin groups. More precisely, if $A_\Gamma$ is an arbitrary Artin group and $A_\Lambda$ is a right-angled Artin group such that their profinite completions are isomorphic, i.e., $\widehat{A_\Gamma}\cong\widehat{A_\Lambda},$ then $\Gamma$ is right-angled and $\Gamma\cong\Lambda$ as labeled graphs. The proof combines the structure of pro-$p$ completions of Artin groups with finite Coxeter quotients to recover the defining graph.

For every integer $n\ge 2$, we introduce a new combinatorial condition PP$(n)$ inspired by paradoxical decompositions of groups. As $n$ increases, the property PP$(n)$ weakens, and the resulting hierarchy interpolates between group-theoretic manifestations of negative curvature and proper proximality. More precisely, we prove that a countable group is properly proximal if and only if it satisfies PP$(n)$ for some $n$. On the other hand, every acylindrically hyperbolic group satisfies PP$(2)$; furthermore, finitely generated PP$(2)$-groups admit a geometric characterization: they are precisely the groups containing strongly quasi-convex, non-cyclic, free subgroups. In particular, we obtain that every countable acylindrically hyperbolic group is properly proximal. Finally, for every $n\in\mathbb N$, we provide examples of finitely generated properly proximal groups that do not satisfy PP$(n)$, thus proving that our hierarchy is indeed infinite.

We determine the full automorphism group of every connected Cayley graph on an almost simple group with a normal connection set. We also characterize exactly when the full automorphism group is generated by right translations, group automorphisms preserving the connection set, and inversion. Our results substantially generalize several results in the literature, including the known determinations of the automorphism groups of derangement graphs, complete transposition graphs and complete alternating group graphs. As a further application, we establish criteria for a finite nonabelian simple group to admit a graphical doubly regular representation and determine exactly which alternating groups do so, correcting claims in the literature.

We prove that random groups in the Gromov density model (under a certain divisibility condition) have property (T) asymptotically almost surely for any density $d > \frac{1}{4}$. This established $\frac{1}{4}$ as the threshold for property (T), since it was shown by Ashcroft that they do not have property (T) for $d < \frac{1}{4}$.

If $U$ is a minimal, diskbusting, finite list of words in a free group $F_n$ of rank $n$ such that the sum of the lengths of words in $U$ is at most $2n+4$, we prove that the natural presentation complex of the Baumslag double of $F_n$ along $U$ virtually contains a $\pi_1$-injective embedded closed hyperbolic surface. This verifies the Tiling Conjecture of Kim and Wilton for this type of lists of words, and in particular, implies that the corresponding Baumslag double contains a hyperbolic surface subgroup.

Kaplansky's zero-divisor conjecture asserts that the group ring of a torsion-free group over a field has no zero divisors. It holds for every group with the unique-product property, so a counterexample can only come from a torsion-free group without unique products. In lectures in 2021, Gardam announced that the torsion-free $\tilde{A}_2$ lattice $\Gamma = \langle a, b \mid a b a^2 b^{-1} a^2 b^{-2}, a b^3 a b^4 a^{-1} b \rangle$ does not have unique products and presented it as a new candidate group: it has property (T), and the known methods for proving the conjecture do not apply to it. To our knowledge, no proof of the announcement has been published. We give a proof checked by the Lean 4 kernel and stated against Mathlib's UniqueProds class. The witness is an explicit pair of finite subsets with $|A| = 32$ and $|B| = 28$ in which each of the 896 products coincides with another product. For 658 products the certificate is an identity in the free group; the other 238 certificates are explicit products of conjugated relators, 970 conjugates in all, checked by free reduction. A homomorphism onto $\mathbb{Z}/42$ shows that each pair $(u,v)$ differs from its partner $(u',v')$ as a pair of group elements, which is all the theorem requires. Together with a homomorphism onto the alternating group $A_4$ it also shows that the listed words are pairwise distinct, so the sets have exactly 32 and 28 elements. The witness and certificates come from an untrusted search program and are re-checked by Lean. The development uses only the axioms propext, Classical.choice and Quot.sound, with no sorry and no native_decide. The mathematical statement is Gardam's. To our knowledge this is the first verification in a proof assistant of a unique-product failure in a torsion-free group; torsion-freeness of $\Gamma$ is taken from Gardam and is not formalized here.

Let $\mathcal{A}_0$ be the set of finitely generated free groups on some countable alphabet and and define $\mathcal{A}_r$ inductively as the union of $\mathcal{A}_{r-1}$ with the set of amalgamated free products $G_1\ast_{\ZZ}G_2$, where $G_1,G_2\in \mathcal{A}_{r-1}$. We show, for all $r$, that every element of $\mathcal{A}_r$ has linear conjugator length, quantifying a result of Larsen and extending a result of Kharlampovich and Myasnikov. This is achieved by a linear conjugator length bound on a family of iterated HNN extensions of free groups. Also, in an appendix, we adapt a proof of A. Wei$\ss$ to show that all Generalized Baumslag-Solitar Groups have linear conjugator length. As a consequence, we obtain that one-relator groups with non-trivial center have linear conjugator length, giving evidence for a conjecture of Bridson, Riley, and Sale.

We prove that every finite Helly complex is isomorphic, at the level of its $1$-skeleton, to the Hellyfication (equivalently, the discrete injective hull) of its combinatorial boundary equipped with the metric induced from the complex. In particular, the boundary-rigidity phenomenon for Helly complexes proved by Blufstein-Chalopin-Chepoi admits a canonical injective-hull interpretation, independent of reconstruction procedures based on dismantling.

We prove that topological isomorphism of procountable groups is a universal analytic equivalence relation, answering a question of Gao, Nies, and Paolini. The same conclusion follows for non-Archimedean Polish groups. More strongly, there is one countable group $H$ for which universality already holds among inverse limits of sequences of surjective endomorphisms of $H$. Thus all the classification complexity can be carried by the bonding maps. We encode permutative equivalence of unconditional basic sequences by integer weights with prescribed symmetries. An extension of Gao, Nies, and Paolini's tree construction recovers bounded weight differences from uniform continuity while allowing these symmetries to act. Przeździecki's almost-full functor transfers the resulting inverse graph systems to groups. Identifying the stage graphs with a single graph makes the passage from weights to bonding endomorphisms continuous.

We prove that the variety generated by all four-element additively idempotent semirings has no finite basis for its identities.

The rank of a semigroup is the cardinality of a smallest generating set. In this paper, we introduce iterated wreath products of transformation semigroups. We determine the rank of iterated wreath products of full transformation monoids and symmetric groups.

In this paper, we explore group invariants arising from combinatorial structures associated with finite groups, including the functional graphs of power maps and the well-studied power graphs. These invariants induce equivalence relations (and hence partitions) on the set of isomorphism classes of finite groups, which we classify from the finest to the coarsest. Surprisingly, all but three of these partitions turn out to coincide; for the subclass of nilpotent groups, all but two coincide. Furthermore, we introduce a broad class of nilpotent groups and show that, within this class, all but one of these partitions agree. Our proofs draw on tools and ideas from Combinatorics and Number Theory, while requiring only basic notions from Group Theory. In particular, we construct a general framework that may prove useful in contexts similar to those considered in this paper. Finally, we propose some open questions that emerge from our results.