Related papers: Burnside groups and $n$-moves for links

Let $G$ be a nontrivial transitive permutation group on a finite set $\Omega$ and recall that an element of $G$ is a derangement if it has no fixed points. Derangements always exist by a classical theorem of Jordan, but there are so-called…

Let $n,d \in \mathbb N$ and $w \in \mathbb F_n$ be non-trivial. We prove that the relatively free group of rank $d$ in the variety defined by the group law $w$ has a largest anabelian finite quotient and estimate its size. Here, a finite…

Let $n,k\in\mathbb{N}$ and let $p_{n}$ denote the $n$th prime number. We define $p_{n}^{(k)}$ recursively as $p_{n}^{(1)}:=p_{n}$ and $p_{n}^{(k)}=p_{p_{n}^{(k-1)}}$, that is, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime. In this note we…

We prove that thick groups (and more generally thick graphs) have trivial Floyd boundary. This shows a wide class of finitely generated groups that are non-relatively hyperbolic have trivial Floyd boundary. In addition to giving new…

The Burnside Problem asks whether a finitely generated group of exponent n is finite. We present a solution for 2-generator groups of prime power exponent. Results of P. Hall and G. Higman extends the finiteness conclusion to groups having…

Let $A$ be a brace of cardinality $p^{n}$ where $p>n+1$ is prime, and let $ann (p^{2})$ be the set of elements of additive order at most $p^{2}$ in this brace. We construct a pre-Lie ring related to the brace $A/ann(p^{2})$. In the case of…

The classes of odd graphs $O_n$ and middle levels graphs $B_n$ form one parameter subclasses of the Kneser graphs and bipartite Kneser graphs respectively. In particular both classes are vertex transitive while resisting definitive…

A famous conjecture of Parkin-Shanks predicts that $p(n)$ is odd with density $1/2$. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with…

We prove an algebraic version of a classical theorem in topology, asserting that an abelian p-group action on a smooth projective variety of positive dimension cannot fix exactly one point. When the group has only two elements, we prove…

We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the…

Using topological notions of translation-like actions introduced by Schneider, we give a positive answer to a geometric version of Burnside problem for locally compact group. The main theorem states that a locally compact group is…

Let $p>3$ be a prime number and let $A$ be a brace whose additive group is a direct sum of cyclic groups of cardinalities larger than $p^{\alpha }$ for some $\alpha $. Suppose that either (i) $A^{\lfloor{\frac {p-1}4}\rfloor}\subseteq pA$…

We classify all groups of order $p^5$ with non-trivial unramified Brauer groups. We show that if $p>3$, then there are precisely $\gcd (p-1,4)+\gcd (p-1,3)+1$ such groups.

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1.…

Let $p$ be an odd prime number. Peyre shows that there is a group $G$ of order $p^{12}$ such that $H_{nr}^3(\bm{C}(G), \bm{Q}/\bm{Z})$ is non-trivial. Using Peyre's method, we are able to prove that the same conclusion is true for some…

This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible…

In the 1950's Milnor defined a family of higher order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received and fruitful study since its inception. In the case that $L$…

In this paper, we prove that if two finite groups G and H have isomorphic Burnside rings, then G and H are the same order type groups, and give an example to show that the Burnside rings of the same order type groups are not necessarily…

Brunnian links have been known for a long time in knot theory, whereas the idea of n-triviality is a recent innovation. We illustrate the relationship between the two concepts with four short theorems.

We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of the $n$-dimensional Euclidean space. For the existence part we…