Related papers: The Mathieu group $M_{12}$ and its pseudogroup ext…

Motivated by the Mathieu conjecture [Ma], the image conjecture [Z3] and the well-known Jacobian conjecture [K] (see also [BCW] and [E]), the notion of Mathieu subspaces as a natural generalization of the notion of ideals has been introduced…

Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative…

Starting from the classical results of Shubnikov and Zamorzayev, computer models of shapes are implemented, which allow to visualize the action of discrete subgroups of continuous topological groups. The action is visualize by performing…

Determining the space of free discrete two generator groups of M\"obius transformations is an old and difficult problem. In this paper we show how to construct large balls of full dimension in this space. To do this, we begin with a marked…

The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic…

We consider a very simple Mealy machine (three states over a two-symbol alphabet), and derive some properties of the semigroup it generates. In particular, this is an infinite, finitely generated semigroup; we show that the growth function…

Loop groups G as families of mappings of the complex manifold M into another complex manifold N preserving marked points $s_0\in M$ and $y_0\in N$ are investigated. Quasi-invariant measures $\mu $ on G relative to dense subgroups $G'$ are…

The purpose of this paper is to study the properties of the irrational-slope Thompson's group $F_\tau$ introduced by Cleary in 1995. We construct presentations, both finite and infinite and we describe its combinatorial structure using…

We consider the group $\mathcal G$ which is the semidirect product of the group of analytic functions with values in ${\mathbb C}^*$ on the circle and the group of analytic diffeomorphisms of the circle that preserve the orientation. Then…

We examine the geometry of the Cayley graph of the lamplighter group with respect to the generating set rising from its interpretation as an automata group due to Grigorchuk and Zuk. We find some metric behavior with respect to this…

We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. To this end we construct groupoids whose Cayley graphs have large girth not just in the usual…

Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are…

We complete the calculation of the Ext algebra of the principal 2-block of the Mathieu group M_11, extending work of Benson-Carlson and of Pawloski - and duplicating work of Generalov.

In this paper we generalize the construction of binary self-orthogonal codes obtained from weakly self-orthogonal designs described by Tonchev in [12] in order to obtain self-orthogonal codes over an arbitrary field. We extend construction…

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a…

We study probability-measure preserving (p.m.p.) actions of finitely generated groups via the graphings they define. We introduce and study the notion of isometric orbit equivalence for p.m.p. actions: two p.m.p. actions are isometric orbit…

We study gauge symmetry in F-theory in light of global aspects. For this, we consider not only a simple (local) group, but also a semi-simple group with Abelian factors. Once we specify the complete gauge group by decomposing the…

Two methods of constructing 2D Toda $\tau$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group…

The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group…

The octonionic root system of the exceptional Lie algebra E_8 has been constructed from the quaternionic roots of F_4 using the Cayley-Dickson doubling procedure where the roots of E_7 correspond to the imaginary octonions. It is proven…