72 papers
This paper investigates the dimension statistics of representations for finite groups, demonstrating that for reductive groups over finite fields and symmetric groups, quantities such as representation dimensions and conjugacy class sizes exhibit asymptotically constant or log-constant behavior as the field size or group rank tends to infinity.
This paper establishes a geometric group theory framework for right quasigroups by introducing graph markings and invariants to characterize their Cayley (di)graphs, thereby proving that all racks are realizable by their full Cayley graphs and providing graph-theoretic characterizations for various algebraic structures.
This paper investigates the sum of the squares of irreducible character degrees not divisible by a prime and its relation to the corresponding quantity in a -Sylow normalizer, thereby proving a recent conjecture by E. Giannelli for the case and several other instances.
This paper establishes the continuity of asymptotic entropy for random walks on wreath products (where is any countable group and is a hyper-FC-central group with a cubic-growth subgroup) by proving the continuity of non-return probabilities and demonstrating that weak continuity of harmonic measures implies entropy continuity, thereby extending known results to new classes of groups including linear and groups.
This paper provides a complete description of the Poisson boundary for wreath products under conditions where lamp configurations stabilize and the projected measure on is Liouville, thereby resolving a long-standing open question regarding the boundary for () with finite first moment measures.
This paper extends the correspondence between automorphism groups and -categorical structures to the locally Roelcke precompact and locally -categorical settings by defining the latter, proving a Ryll-Nardzewski theorem, characterizing the associated groups via isometric actions, and establishing that bi-interpretability of structures is equivalent to the isomorphism of their automorphism groups.
This paper establishes that sufficiently large topologically generating sets in connected compact, amenable, and reductive algebraic groups are necessarily redundant, providing quantitative bounds linked to finite simple groups of Lie type and demonstrating that these findings partially resolve Gelander's conjectures by showing they follow from the Wiegold conjecture.
This paper investigates the structural and combinatorial properties of Bi-Cayley graphs over cyclic groups of order , establishing that they are connected, biregular, have girth three and diameter five, while also extending these findings to arbitrary finite groups under specific conditions.
This paper establishes that for a compact Lie group and sufficiently large rank , the dynamics of the automorphism group acting on the representation space exhibit algebraic rigidity, where orbit closures and invariant probability measures are algebraic in nature, analogous to Ratner's theorems.
This paper proves that the smallest group satisfying Guido Zappa's 1962 question regarding cosets of Sylow -subgroups containing only elements of -power order cannot be an alternating simple group for any prime .
This paper introduces the concepts of "siblings" and "twins" to analyze the difficulty of distinguishing non-isomorphic p-groups, ultimately applying these insights to develop an effective algorithm for identifying all 10,494,213 groups of order $2^9$.
This paper establishes a comprehensive theory of multiary graded polyadic algebras by extending classical group-graded algebras to higher-arity structures, introducing multiary group gradings, and uncovering unique phenomena such as higher power gradings and specific arity compatibility constraints through results like quantization rules and a First Isomorphism Theorem.
This paper investigates the existence of invariant transversals for normal subgroups and provides counterexamples to a conjecture in the specific case where the subgroup is abelian and the group is finite.
This paper classifies free graph braid groups using cubical structures and demonstrates that, under specific conditions, the large-scale geometry of graph 2-braid groups is determined by the union of maximal product subcomplexes, yielding new examples of groups that are and are not quasi-isometric to right-angled Artin groups.
This paper presents a polynomial-time algorithm for testing the isomorphism of finite groups that are extensions of abelian groups by -generated groups (including abelian-by-cyclic and abelian-by-simple cases), achieved through a novel method for computing the unit group of a finite ring.
This paper establishes an equivalence between a conjecture by Neumann and Praeger concerning Kronecker classes in algebraic number fields and a separate conjecture regarding cliques in derangement graphs within combinatorics.
This paper resolves the isomorphism problem for reduced finitary power monoids of cancellative monoids by proving that if one monoid is torsion, the power monoids are isomorphic if and only if the base monoids are, thereby confirming the result for torsion groups while leaving the general group case open.
This paper confirms the Bader-Kropholler-Vankov conjecture by proving that homological filling functions over a ring with a discrete norm are quasi-isometry invariants for all groups of type , utilizing a novel technique of equipping free chain complexes with geometric structures to extend cellular constructions to the algebraic setting.
This paper investigates higher property T by providing new operator-algebraic characterizations and exploring its relationship with cohomological, rigidity, and geometric phenomena in lattices of semisimple Lie groups below their real rank, culminating in a unifying conjectural framework.
This paper investigates the arithmetic properties of factorizations in finitely-presented monoids by analyzing how presentation relations influence factorization behavior, constructing non-commutative fully elastic monoids, and proving that finitely-presented cancellative normalizing monoids satisfy the Structure Theorem for Unions.