26,110 papers in this slice of arXiv.
Emily Gullerud, Peter Webb
We introduce synthetic buildings for each finite group G and prime p. These are G-simplicial complexes, among which are the p
M. H. Shahzamanian
The determinant of a finite semigroup is closely related to the Frobenius property of its corresponding semigroup algebra, and it plays a significant role in coding theory applications. In this paper, we study determinants of twisted contracted semigroup algebras over finite fields. We extend Steinberg's determinant theory to this setting by establishing a Frobenius criterion for twisted contracted semigroup algebras over finite fields. We also develop a structural theory for two-generated twisted contracted monoid algebras. In particular, we identify classes for which the determinant of the associated matrix depends only on its zero--nonzero pattern, and is therefore independent of the values of the cocycle. Consequently, it suffices to verify the determinant condition for the trivial character, from which the corresponding result for all characters follows. Under suitable assumptions, these semigroups are completely determined by a small set of defining identities. Motivated by the character-theoretic construction of homogeneous weights on finite Frobenius rings, we construct homogeneous weights for Frobenius twisted contracted semigroup algebras and investigate the non-Frobenius case. In particular, we characterize the principal ideals on which the character-average weight satisfies the averaging property of homogeneous weights and thereby extend the classical notion of homogeneous weight to a prescribed family of principal ideals.
Andrey V. Vasil'ev
In this short note we address the problems of characterization of simple and related groups by various arithmetic invariants. We come with some uniform way to think about such sort of questions and discuss what has been already done and what we still do not know but wish to know in this field.
Ruslan Skuratovskii
The size of minimal systems of generators and these systems themselves for groups ESLn[Z] and ESLn(Fp)
Jonathan Feigert, Daniel Herden, Marcos Mazari-Armida
We study the abstract elementary class of acts with pure embeddings. In particular, we show that stability and superstability in this class can be characterized in terms of the monoid S being LO (for every s,t∈S, we have that s∈St or t∈Ss) and weakly noetherian (every ideal is finitely generated), respectively. Moreover, under mild set-theoretic assumptions, we characterize the stability spectrum via the minimal cardinality of a generating set for every ideal. As an application, we obtain a Baer-like criterion for pure injective acts when S is LO. We use this to provide an alternative proof that the class of acts has enough pure injectives when S is LO.
Benjamin Marsh
Let k≥3, n≥1, and let H_k,n=(C_k,n) be the group generated by the standard k pile perfect shuffle and cyclic pile permutation on a deck of kn cards. We prove that Hk,n is 2-transitive whenever n is not a power of k. Residual commutators give translations supported on two pile labels, and a strongly connected digit digraph propagates these translations throughout the deck, a separate argument resolves the antipodal support case. We then combine this result with fixed point ratio bounds for primitive groups and explicit boundary calculations to determine Hk,n for all k and n. If n=kf, then Hk,n≅Ck≀Cf+1. If k=4 and n=2⋅4j, then H₄,n≅(2j+3,2). In every other case, Hk,n is (kn) or (kn), according to the parity of its generators. This proves Conjecture 1.10 of Amarra, Morgan and Praeger and Conjecture 5.1 of Xia, Zhang, Zhang and Zhu. More generally, we classify (P,n) for every pile group P containing Ck, and obtain the odd k part of their Conjecture 5.2.
Liam May, Yong Yang
We find a generating function for the number of ordered commuting pairs of elements in a symplectic Lie algebra over a finite field of characteristic 2, following the analogous work done in odd characteristic in Fulman and Guralnick (arXiv:1611.05499).
Liam May, Yong Yang
We determine the connected components of the divisibility graph of several families of finite groups of Lie type in defining characteristic 2, extending the result of Abdolghafourian, Iranmanesh, and Niemeyer (arXiv:1612.04410), who treated odd defining characteristic. We show that there is a distinguished component containing all nonidentity unipotent class sizes and that every other component is either an isolated vertex or an explicitly described two-vertex component. We also determine the isolated vertices outside the distinguished component. Additionally, we prove in Appendix A that the divisibility graph of a Frobenius group has exactly two components.
Sam K. Miller
Tornehave and Yalçin proved that the tom Dieck homomorphism, which sends a virtual real representation to a unit of the Burnside ring, is surjective for any p-group S. We prove that this homomorphism, and the sign homomorphism it factors through, remain surjective when restricted to fusion-stable subgroups associated to a saturated fusion system on S. As a corollary, we close the main question posed by Mazza--Miller in arXiv:2508.07404 by showing that given a field k of positive characteristic, the Lefschetz homomorphism from the Picard group of the bounded homotopy category of p-permutation modules to the unit group of its Grothendieck ring is surjective for all finite groups if and only if k=F2.
Peter Haïssinsky, Yusheng Luo
We study the quasisymmetric classification of limit sets of Kleinian groups and obtain a characterization of those geometrically finite limit sets that are quasisymmetrically universal. This allows us to obtain a quasi-isometric classification of finitely generated Kleinian groups. We also obtain a classification of virtually topologically rigid geometrically finite Kleinian groups.
Ilya Kapovich
For every n≥8, we construct explicit families of generating pairs of GL(n,Z) and SL(n,Z) representing countably infinitely many Nielsen equivalence classes. These pairs arise as the homology images of explicit torsion generating pairs in Out(Fn) and its index-two subgroup SOut(Fn), yielding countably infinitely many Nielsen classes of generating pairs in both outer groups. Within each constructed outer family, all pairs become Nielsen equivalent after one stabilization, obtained by adjoining a trivial coordinate. We derive a relation-module obstruction to Nielsen equivalence of once-stabilized generating tuples. Evans's non-cancellation examples show that this obstruction is nontrivial and can distinguish Nielsen classes of once-stabilized generating tuples. We also record a quotient relation-module refinement for possible future applications.
Leonardo Dinamarca, Maximiliano Escayola, Sang-hyun Kim +1
We investigate acylindrically hyperbolic groups acting on the circle by piecewise linear homeomorphisms. We prove that a finitely generated acylindrically hyperbolic group acting faithfully by piecewise linear homeomorphisms of the circle is virtually free, is virtually a closed hyperbolic surface group, or virtually splits over a two-ended subgroup; as a consequence, we prove a general structural result for Gromov hyperbolic groups of piecewise linear homeomorphisms. Along the way, we show that a right-angled Artin subgroup of piecewise linear homeomorphisms of the circle is either free or abelian, generalizing a result of Bleak and Salazar-Diaz. We also show that the fundamental group of a finite volume hyperbolic n--manifold cannot act faithfully on the circle by piecewise linear homeomorphisms, provided that n≥3, complementing 2--dimensional examples of Ghys and Minakawa.
Eli Glasner
We prove that a Hausdorff topological group is tame if and only if it is precompact.
Mario Klisse
Given a countable discrete group equipped with a proper length function, one can construct a natural spectral triple on its reduced group C∗-algebra. A well-studied question in non-commutative metric geometry is whether the Connes pseudo-metric associated with such a triple recovers the weak∗-topology on the state space, thereby yielding a compact quantum metric space in the sense of Rieffel. While this metric property is known to hold for several classes of groups - including those of polynomial growth and word-hyperbolic groups - it was widely expected that not every word-length function induces a compact quantum metric space. Despite this, no explicit counterexample has been identified to date. In this note, we provide the first family of counterexamples by proving that for every integer d≥2 the canonical spectral triple of the Lamplighter group (Z/2Z)≀Fd, equipped with the word-length function associated with a finite symmetric generating set, fails to be a spectral metric space.
Charles Eaton, Florian Eisele, Radha Kessar +2
We determine the Morita equivalence classes of 2-blocks with quaternion defect groups of arbitrary 2-power order, thereby completing the proof of Donovan's conjecture for blocks of tame representation type.
Richie Sater
Let d(G) denote the least size of a generating set of a finite group G. We prove that if G has a family H of subgroups such that d(H)≤d for every H∈H and gcd{∣G:H∣:H∈H}=1, then d(G)≤d+1. This gives an affirmative answer to Kourovka Problem 21.87. The proof reduces a minimal counterexample to a critical crown-based power with nonabelian socle. An exact crown multiplicity formula and a uniform lower bound for conditional generation give a lower bound for the number of crown factors. A subgroup containing a Sylow 2-subgroup gives the contradictory upper bound, via a pointwise centralizer estimate for Sylow 2-subgroups of finite simple groups.
Maxim Ivanov
We introduce C-lifting functors, which axiomatize lifting properties of non-abelian tensor and exterior products with respect to prescribed classes of group extensions. For a homomorphism f:Γ→G, we associate to every C-lifting functor F a relative quotient Ff(G). This quotient maps epimorphically onto the subgroup determined by F in every f-extension belonging to C. We show that the construction is functorial in f and that, for an f-extension p:G→G, it gives a morphism of natural exact sequences. In degree two this yields an epimorphism f∗H2(Γ;Z)H2(G;Z)⟶kerp∩[G,G]. Applying the construction to iterated tensor and exterior powers, we obtain relative Schur-Baer theorems for the lower central and derived series. We also compare the relative tensor and exterior squares, relate the exterior construction to the relative Schur multiplier H2(G,Γ;Z), and derive applications to orderability. As a consequence, we prove that a virtual knot group is left-orderable if and only if it is circularly orderable.
Paul Flavell
Glauberman's characteristic p-functor K∞ has a number of remarkable properties. For example it controls p-transfer in any finite group provided p≥5. These notes are the author's attempt to understand and simplify this work.
Yifan Jing, Shukun Wu
We study two Kakeya set problems in compact semisimple Lie groups. In the first, motivated by the group structure, a Kakeya set is required to contain a left coset of every closed one-dimensional torus. For a compact semisimple Lie group G of dimension d and rank r, we determine the optimal Minkowski dimension for this problem, proving that it is exactly (d+r)/2. The upper bound is obtained from a Lie-theoretic construction associated with a Chevalley involution, while the lower bound combines incidence geometry with geometry of numbers. We also consider a local Kakeya set problem, closer to the classical harmonic-analytic formulation, in which one requires a fixed-length one-parameter arc in every direction. For this problem we obtain lower bound (d+1)/2 and (d+2)/2 for odd and even r, and upper bound (d+r)/2. We conjecture that the upper bound should be sharp.
Yannik Höll, Friedrich Martin Schneider
Given a non-atomic standard probability space (Ω,μ), we exhibit examples of non-amenable closed topological subgroups of Aut(Ω,μ) whose natural near-action on (Ω,μ) is whirly. This answers a 2010 question by Pestov in the negative. The argument proceeds via constructing Gaussian near-actions from topologically faithful unitary representations of non-amenable whirly Polish groups.