Related papers: On 3 and 4 dimensional regular solids, Part 1: The…

We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure three-braid we give effective upper and lower bounds for these invariants. This is done in terms of a natural syllable decomposition of the…

Palindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups $G$ has infinite palindromic width, provided that $G$ is not the free product of two cyclic groups…

The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups. We prove that all automorphisms are tame and the automorphism group is…

Let $S$ be the numerical semigroup generated by three consecutive numbers $a,a+1,a+2$, where $a\in\mathbb{N}$, $a\geq 3$. We describe the elements of $S$ whose factorizations have all the same length, as well as the set of factorizations of…

Linear second order recursive sequences with arbitrary initial conditions are studied. For sequences with the same parameters a ring and a group is attached, and isomorphisms and homomorphisms are established for related parameters. In the…

A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always…

We show that for any given n, there exists a sequence of words a_k in the generators sigma_1, ... sigma_{n-1} of the braid group B_n, representing the identity element of B_n, such that the number of braid relations of the form sigma_i…

In this paper we study algebras of modular forms on unitary groups of signature $(n,1)$. We give a necessary and sufficient condition for an algebra of unitary modular forms to be free in terms of the modular Jacobian. As a corollary we…

We show that the minimum number of vertices of a simplicial complex with fundamental group $\mathbb{Z}^{n}$ is at most $O(n)$ and at least $\Omega(n^{3/4})$. For the upper bound, we use a result on orthogonal 1-factorizations of $K_{2n}$.…

We write out some sequences of linear maps of vector spaces with fixed bases. Each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex - a triangulation of a manifold. If…

Suppose $G$ is a simple group. For any nontrivial elements $g$ and $h$, $g$ can be written as a finite product of conjugates of $h$ or the inverse of $h$. G is called uniformly simple if the length of such an expression is uniformly…

In the present paper, we define an invariant of free links valued in a free product of some copies of $\mathbb{Z}_{2}$. In \cite{Ma2} the second named author constructed a connection between classical braid group and group presentation…

We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and…

This note provides an alternate account of Calegari's rationality theorem for stable commutator length in free groups.

We verify that every alternating group of degree at most one quadrillion is invariably generated by an element of prime order together with an element of prime power order.

We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their abelianization (or equivalently, their integer homology class. We derive an explicit generating function, and a limiting…

Long before we learn to construct the field of rational numbers (out of the ring of integers) at university, we learn how to calculate with fractions at school. When it comes to "numbers", we are used to a commutative multiplication, for…

We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine…

We consider some questions about formal languages that arise when inverses of letters, words and languages are defined. The reduced representation of a language over the free monoid is its unique equivalent representation in the free group.…

We construct words with small image in a given finite alternating or unimodular group. This shows that word width in these groups is unbounded in general.