Related papers: Residual nilpotence and ordering in one-relator gr…

We introduce a version of the Alexander polynomial for singular knots and tangles and show how it can be strengthened considerably by introducing a perturbation. For singular long knots, we also prove that our Alexander polynomial agrees…

A non-trivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for…

The following criterion is proved in this paper. If the Alexander polynomial of a knot $K\subset S^3$ has a zero of odd order on the complex unit circle, then there exists a continuous family of irreducible representations…

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we…

A class of one-relator groups such that every group in the class is determined by a triple of integers and is an HNN-extension of some Baumslag -- Solitar group is considered. A criterion for two groups in this class to be isomorphic and…

We construct countable groups $G$ with the following new degree of W*-superrigidity: if $L(G)$ is virtually isomorphic, in the sense of admitting a bifinite bimodule, with any other group von Neumann algebra $L(\Lambda)$, then the groups…

It is shown that finite groups in which the order of the product of every pair of elements of co-prime order is the product of the orders, is nilpotent.

Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.

Let $G$ be a finitely generated group with polynomial growth, and let $\om$ be a weight, i.e. a sub-multiplicative function on $G$ with positive values. We study when the weighted group algebra $\ell^1(G,\om)$ is isomorphic to an operator…

Let $G$ be a finite group admitting a coprime automorphism $\alpha$. Let $J_G(\alpha)$ denote the set of all commutators $[x,\alpha]$, where $x$ belongs to an $\alpha$-invariant Sylow subgroup of $G$. We show that $[G,\alpha]$ is soluble or…

A group $G$ is called root graded if it has a family of subgroups $G_\alpha$ indexed by roots from a root system $\Phi$ satisfying natural conditions similar to Chevalley groups over commutative unital rings. For any such group there is a…

We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.

Consider a group word w in n letters. For a compact group G, w induces a map G^n \rightarrow G$ and thus a pushforward measure {\mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and…

Let m, n be positive integers, v a multilinear commutator word and w = v^m. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are…

We show that the intersection of the rational derived series of a one-relator group is rationally perfect and is normally generated by a single element. As a corollary, we characterise precisely when a one-relator group is residually…

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot…

Let $G$ be a group generated by a set $X$. It is well known and easy to check that \[ [g_1, g_2, \dots ,g_n] = 1 \mbox{ for all } g_i \in G \qquad \iff \qquad [x_1, x_2, \dots , x_n] =1 \mbox{ for all } x_i \in X. \] Let $L$ be a Lie…

Let $K$ be a subgroup of a finite group $G$. The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. Assume that $Pr(K,G)\geq\epsilon$ for some fixed $\epsilon>0$. We show that there is a normal…

First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field.…

The definition of \NP\ requires, for each member language~$L$, a polynomial-time checking relation~$R$ and a constant~$k$ such that $w \in L \iff \exists y\,(|y| \leq |w|^k \wedge R(w,y))$. We show that this biconditional instantiates, for…