72 papers
This paper establishes a folklore theorem characterizing the embeddability of partial groups in groups via unique parenthesization of words, investigates examples of non-embeddable cases, and proves that a partial groupoid embeds in a groupoid if and only if its reduction embeds in a group.
This paper establishes that for any finite group possessing a faithful irreducible representation, every finite supergroup contains an irreducible representation whose restriction to is faithful and center-preserving, thereby proving that such representations always exist in the induction of a faithful representation of .
This paper establishes the decidability of the word problem for various classes of just infinite groups with recursively enumerable relations using direct methods independent of the Wilson–Grigorchuk classification, while also constructing specific counterexamples of locally finite groups where the problem is undecidable for certain presentations.
This paper establishes a cohomological criterion for decomposing Borel graphs, analogous to Dunwoody's work on group accessibility, and applies it to prove that Borel graphs with uniformly bounded degrees and cohomological dimension one are Lipschitz equivalent to acyclic graphs, thereby providing a new proof of a result by Chen et al. regarding graphs with tree-like components.
This paper establishes that outer automorphisms of virtually special groups exhibit either polynomial or exponential growth with algebraic integer stretch factors, constructs a Nielsen-Thurston-like decomposition for coarse-median preserving automorphisms, and proves that their outer automorphism groups satisfy the Tits alternative, are boundary amenable, and have finite virtual cohomological dimension.
This paper investigates the structural and spectral properties of Engel and co-Engel graphs associated with finite groups, establishing that the undirected Engel graph does not uniquely determine its directed counterpart, characterizing isolated vertices via the Fitting subgroup, and computing topological and spectral invariants to classify non-Engel groups with specific graph-theoretic constraints.
This paper introduces an antichain condition for non- subgroups in infinite groups and proves that, within the class of generalized radical groups, this condition is equivalent to the restricted chain condition (RCC), thereby establishing minimax-type dichotomies for various subgroup properties such as normality, permutability, and pronormality.
This paper establishes that an affine transformation in is a product of two coninvolutions if and only if its linear part is -reversible, while also characterizing products of three coninvolutions and proving that every transformation with a linear part of unit determinant is a product of at most four coninvolutions.
This paper investigates the local-triviality dimensions of actions on -algebras within noncommutative Borsuk-Ulam theory, demonstrating that free actions do not necessarily possess finite weak local-triviality dimensions and that these invariants can exhibit discontinuity or exceed fiber values in continuous fields, while establishing conditions for upper semicontinuity through examples involving noncommutative tori and spheres.
This paper proves that the Quillen posets of -extensions of simple unitary groups have non-zero homology in the largest possible dimension, thereby establishing the 1992 Aschbacher-Smith conjecture and confirming Quillen's conjecture for odd primes.
This paper establishes that for most parameter sets, the symmetry group of Markoff-type equations acts transitively on the majority of solutions modulo for a density one set of primes, with specific applications proving near-complete transitivity results for the -classification conjecture in and for solutions arising from generalized cluster algebras.
This paper establishes that both geometric and homological finiteness properties of group pairs remain invariant under a suitably defined notion of quasi-isometry for group pairs.
This paper investigates the poset of subgroup classes in finite groups sharing the same set of element orders, characterizing the groups for which this poset is a chain or specifically , and analyzing its lattice properties, including distributivity and modularity, for cyclic and dihedral groups.
This paper investigates finiteness conditions on skew braces, specifically -skew braces with additive groups, by establishing structural analogs to finite conjugacy and demonstrating how these properties characterize infinite set-theoretic solutions of the Yang-Baxter equation that behave similarly to finite ones.
This paper proposes a framework for Haar-type measures on topological quasigroups by introducing quasi-invariant measures with modular cocycles, demonstrating how Moufang-type identities constrain these cocycles to suggest that Kunen's theorem reflects the collapse of modular defects into a loop structure.
This paper investigates which finitely generated groups admit proper actions on finite products of locally finite trees, presenting evidence that hyperbolic surface groups possess such actions and providing an explicit embedding of the genus 2 surface group into for any prime .
This paper demonstrates that in the general framework of -boundaries, the boundary of an infinitely ended group admitting a splitting over finite subgroups is canonically homeomorphic to the dense amalgam of the limit sets of its factor subgroups.
This paper establishes the theoretical framework for solving axial symmetric Navier-Stokes equations in a cylindrical topology by constructing a complete basis of harmonic 1-forms comprising Beltrami, anti-Beltrami, and closed components, thereby reducing the problem to a hierarchy of quadratic relations suitable for future optimization via Physics-Informed Neural Networks.
This paper provides combinatorial characterizations of virtually torsion-free and virtually free groups by utilizing canonical graph decomposition theory to establish necessary and sufficient conditions involving the structural properties of -local covers and their associated tree-decompositions.
This paper establishes that for any integer and any greater than the unique positive root of a specific polynomial, the restricted -fold sumset of a subset in a finite abelian group of sufficiently large odd order equals the entire group whenever , thereby generalizing previous results on cyclic groups and identifying as the optimal asymptotic density threshold.