Related papers: Combinatorial descriptions of multi-vertex 2-compl…

We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric…

We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a…

We introduce polytopal cell complexes associated with partial acyclic orientations of a simple graph, which generalize acyclic orientations. Using the theory of cellular resolutions, two of these polytopal cell complexes are observed to…

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the graph whose vertex set is $G$, and two elements in $G$ are adjacent if one of them is a power of the other. The purpose of this paper is twofold. First, we find the complexity of…

We introduce L-presentations: group presentations given by a generating set, a set of relations and a set of substitution rules on the generating set producing more relations. We first study in full generality the structure of finitely…

A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators $x_1, ..., x_n$ and the cyclic relations: $x_{i_k}x_{i_{k-1}} ... x_{i_1} = x_{i_{k-1}} ... x_{i_1} x_{i_k} = ... =…

Double groupoids are a type of higher groupoid structure that can arise when one has two distinct groupoid products on the same set of arrows. A particularly important example of such structures is the irrational torus and, more generally,…

This is an announcement of some of the results obtained as a part of the second author's Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is…

Burau representation of the Artin braid group remains as one of the very important representations for the braid group. Partly, because of its connections to the Alexander polynomial which is one of the first and most useful invariants for…

Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They…

We give conditions on a presentation of a group, which imply that its Cayley complex is simplicial and the flag complex of the Cayley complex is systolic. We then apply this to Garside groups and Artin groups. We give a classification of…

We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with…

This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…

This is an expository book on unitary representations of topological groups, and of several dual spaces, which are spaces of such representations up to some equivalence. The most important notions are defined for topological groups, but a…

A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the…

Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will…

We present an algorithm which given a presentation of a group $G$ without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant ${\kappa}$, outputs a collection of tracks in an…

New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…

We establish a new group-theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras of ADE types in the same spirit of our new approach to the McKay correspondence. Our vertex operator…

Construction of representations of braid group generators from $N$-state vertex models provide an elegant route to study knot and link invariants. Using such a braid group representation, an algebraic formula for the link invariants was put…