Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing element $x$. A famous theorem B. H. Neumann states that if is in which all classes are finite with bounded size, then derived $G'$ finite. We establish following result. Let $n$ be positive integer and $K$ subgroup such $|x^G|\leq n$ each $x\in K$. $H=\langle K^G\rangle$ normal closure $K$. Then order $H'$ $n$-bounded. ...

This paper defines Artin groups with m-gon structure and proves that for m > 3 in this class the problems of equality conjugacy words are solvable.

Friedland and Milnor [FM] have shown that from a dynamical point of view the polynomial diffeomorphisms of C2 fall naturally into two classes. The first class consists of diffeomorphisms with simple dynamics. The diffeomorphisms in this class have periodic points of at most finitely many periods and topological entropy zero. The second class contains the well-known Henon map f(x, y) = (y, i-ax ...

LetM be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π1(M) is efficient with respect to the JSJ decomposition of M . We go on to prove that π1(M) is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if M is a graph manifold then π1(M) is conjugacy separable. A group G i...

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finiteorder elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type Fn inside mapping class groups, Aut(Fk), and Out(Fk) which have infin...

An analogue of the correspondence between GL(k)-conjugacy classes of matricial polynomials and line bundles is given for K-conjugacy classes, where K ⊂ GL(k) is one of the following: maximal parabolic, maximal torus, GL(k − 1) embedded diagonally. The generalised Legendre transform construction of hyperkähler metrics is studied further, showing that many known hyperkähler metrics (including the...

I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures D of K-contact type one obtains a configuration of Sasaki cones called a bouquet such...

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results...