Related papers: Effective Twisted Conjugacy Separability of Nilpot…

We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics,…

We consider twisted conjugacy classes of continuous automorphisms $\varphi$ of a Lie group $G$. We obtain a necessary and sufficient condition on $\varphi$ for its Reidemeister number, the number of twisted conjugacy classes, to be infinite…

Coclass theory has been a highly successful approach towards the investigation and classification of finite nilpotent groups. Here we suggest a similar approach for finite nilpotent semigroups. This differs from the group theory setting in…

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq \epsilon >…

A new general formula for the number of conjugacy classes of subgroups of given index in a finitely generated group is obtained.

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.

Reidemeister numbers of group automorphisms encode the number of twisted conjugacy classes of groups and might yield information about self-maps of spaces related to the given objects. Here we address a question posed by Gon\c{c}alves and…

This thesis deals with the conjugacy problem in groups and its twisted variants. We analyze recent results by Bogopolski, Martino, Maslakova and Ventura on the twisted conjugacy problem in free groups and its implication for the conjugacy…

We study the strong, weak, and special amalgamation bases in the varieties of nilpotent groups of class two and exponent n, where n is odd. The main result is a characterization of the special amalgamation bases for these varieties. We also…

In this paper, we discuss twisted Alexander polynomials of a knot for group extensions of a finite group in two directions. Firstly, we provide a mod $p$ formula for the twisted Alexander polynomial of a knot in the $3$-sphere associated…

We prove that the fundamental group of any Seifert 3-manifold is conjugacy separable. That is, conjugates may be distinguished in finite quotients or, equivalently, conjugacy classes are closed in the pro-finite topology.

Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the…

This paper has two main parts. In the first part we develop an elementary coordinatization for any nilpotent group $G$ taking exponents in a binomial principal ideal domain (PID) $A$. In case that the additive group $A^+$ of $A$ is finitely…

Let $G$ be a nilpotent group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy classes of $G$ with the property that $|a^G|=|b^G|=p$, where $p$ is an odd prime…

We show that a finitely generated soluble group is virtually nilpotent if and only if the diameter of its finite coset spaces admits a uniform polynomial lower bound in terms of their size. We obtain the same conclusion for certain finitely…

It is proved that generalized free product of two finite p-groups is a conjugacy p-separable group if and only if it is residually finite p-groups. This result is then applied to establish some sufficient conditions for conjugacy…

We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of…

In this paper, we determine the structure of the nilpotent multipliers of all pairs $(G,N)$ of finitely generated abelian groups where $N$ admits a complement in $G$. Moreover, some inequalities for the nilpotent multipliers of pairs of…

We present a structural description of finite nilpotent groups of class at most $2$ using a specified number of subdirect and central products of $2$-generated such groups. As a corollary, we show that all of these groups are isomorphic to…

A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable…