Related papers: Burnside groups and $n$-moves for links
Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n \gg 1$ is solved in the affirmative.…
In a pair of recent papers (one to appear and one forthcoming), the author develops a general version of small cancellation theory applicable in higher dimensions, and then applies this theory to the Burnside groups of sufficiently large…
Let $p_{1}$, ..., $p_{k}$ be the first $k$ odd primes in succession. Let $n$ be an even integer such that $n > p_{k}$. We conjecture that if none of $n - p_{1}$, ..., $n - p_{k}$ are prime, then at least one of them has a prime factor which…
It is well known that many famous Burnside-type problems have positive solutions for PI-groups and PI-algebras. In the present article we also consider various Burnside-type problems for PI-groups and PI-representations of groups.
A. Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito…
It is a simple fact that a group has a trivial automorphism group if and only if it is of order $1$ or $2$. We prove that the same holds for certain families of skew braces, and given any odd prime $p$, we construct a skew brace of order…
In this paper we study geometric versions of Burnside's Problem and the von Neumann Conjecture. This is done by considering the notion of a translation-like action. Translation-like actions were introduced by Kevin Whyte as a geometric…
We introduce a new class of transitive permutation groups which properly contains the automorphism groups of vertex-transitive graphs and digraphs. We then give a sufficient condition for a quotient of this family to remain in the family,…
We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between…
We introduce a model for random groups in varieties of $n$-periodic groups as $n$-periodic quotients of triangular random groups. We show that for an explicit $d_{\mathrm{crit}}\in(1/3,1/2)$, for densities $d\in(1/3,d_{\mathrm{crit}})$ and…
It is a natural question to ask whether two links are equivalent by the following moves -- parallel parts of a link are changed to k-times half-twisted parts and if they are, how many moves are needed to go from one link to the other. In…
We construct novel examples of finitely generated groups that exhibit seemingly-contradicting probabilistic behaviors with respect to Burnside laws. We construct a finitely generated group that satisfies a Burnside law, namely a law of the…
We study the notion of twisted conjugacy separability (essentially introduced in our previous paper for a proof of twisted version of Burnside-Frobenius theorem) and some related properties. We give examples of groups with and without this…
The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we…
We develop a version of small cancellation theory in the variety of Burnside groups. More precisely, we show that there exists a critical exponent $n_0$ such that for every odd integer $n\geq n_0$, the well-known classical $C'(1/6)$-small…
We study paths in the p-Bratteli diagram associated with hook partitions, where p is an odd prime. By comparing blocks along a path, we define inversions and descents. We prove that the sign balance derived from inversions vanishes at every…
We construct first examples of non-trivial groups without non-cyclic free subgroups whose reduced $C^\ast$-algebra is simple and has unique trace. This answers a question of de la Harpe. Both torsion and torsion free examples are provided.…
The homotopy trivializing number, \(n_h(L)\), and the Delta homotopy trivializing number, \(n_\Delta(L)\), are invariants of the link homotopy class of \(L\) which count how many crossing changes or Delta moves are needed to reduce that…
For any pair of integers $m$ and $n$ such that $3<m<n$, we provide an infinite family of links, where each link in the family has a locally minimal $n$-bridge position and a globally minimal $m$-bridge position. We accomplish this by…
In this note, we define the Burnside ring of a monoid, generalizing the construction for groups. After giving foundational definitions, we characterize transitive M-sets and their automorphisms, then prove a structure theorem for a broad…