Related papers: Generators of Bieberbach groups with 2-generated h…

A classical theorem of Forster asserts that a finite module $M$ of rank $\leq n$ over a Noetherian ring of Krull dimension $d$ can be generated by $n + d$ elements. We prove a generalization of this result, with "module" replaced by…

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…

Cyclic sieving is a well-known phenomenon where certain interesting polynomials, especially $q$-analogues, have useful interpretations related to actions and representations of the cyclic group. We propose a definition of sieving for an…

Little is known about the generators system of the higher dimensional Picard modular groups. In this paper, we prove that the higher dimensional Eisenstein--Picard modular group $\mathbf{PU}(3,1;\mathbb{Z}[\omega_3])$ in three complex…

We give an explicit and essentially minimal list of defining relations of a Nichols algebra of diagonal type with finite root system. This list contains the well-known quantum Serre relations but also many new variations. A conjecture by…

This survey paper is devoted to Riemannian manifolds with special holonomy. To any Riemannian manifold of dimension n is associated a closed subgroup of SO(n), the holonomy group; this is one of the most basic invariants of the metric. A…

Combinatorial Hantzsche-Wendt groups were introduced by W. Craig and P.A. Linnell. Every such a group $G_n$, where $n$ is a natural number, encodes the holonomy action of any $n+1$-dimensional Hantzsche-Wendt manifold. $G_2$ is the…

The goal of this thesis is to define a 2-dimensional version of abelian categories, where symmetric 2-groups play the role that abelian groups played in 1-dimensional algebra. Abelian and 2-abelian groupoid enriched categories are defined…

The paper extends Birkhoff's theorem on doubly stochastic matrices to some countable families of discrete probability spaces with nonempty intersections. We join every two elements lying in the same probability space by an edge and…

We describe the homotopy classes of 2 by 2 periodic simple (=non-degenerate) matrices with various symmetries. This turns out to be an elementary exercise in the homotopy of closed curves in three dimensions. The matrices represent gapped…

The holonomy group of an (n+2)-dimensional simply-connected, indecomposable but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic group $(\mathbb{R} \times SO(n))\ltimes \mathbb{R}^n$. The main ingredient of such a…

W.Haebich (Bull. Austral. Math. Soc., 7, 1972, 279-296) presented a formula for the Schur multiplier of a regular product of groups. In this paper first, it is shown that the Baer-invariant of a nilpotent product of groups with respect to…

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-space, take the quotient of the remaining space by a discrete group, and find generators for the…

A set of ring generators for the Hecke algebra of the Gel'fand pair $(S_{2n},B_n)$, where $B_n$ is the hyperoctahedral subgroup of the symmetric group $S_{2n}$ is presented. Various corollaries are given. A conjecture of Sho Matsumoto is…

To a set $\mathcal{B}$ of 4-subsets of a set $\Omega$ of size $n$ we introduce an invariant called the `hole stabilizer' which generalises a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Loyd's…

The aim of the present paper is to obtain a classification of all the irreducible modular representations of the symmetric group on $n$ letters of dimension at most $n^3$, including dimension formulae. This is achieved by improving an idea,…

This article provides a decidable criterion for when a subgroup of Out(Fr) generated by two Dehn twists consists entirely of polynomially growing elements, answering an earlier question of the author.

Let Gamma be a group generated by two positive multi-twists. We give some sufficient conditions for Gamma to be free or have no `unexpectedly reducible' elements. For a group Gamma generated by two Dehn twists, we classify the elements in…

The 2-girth of a 2-dimensional simplicial complex $X$ is the minimum size of a non-zero 2-cycle in $H_2(X, \mathbb{Z}/2)$. We consider the maximum possible girth of a complex with $n$ vertices and $m$ 2-faces. If $m = n^{2 + \alpha}$ for…

A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response…