ON THE PROBLEM OF WEAK POWER CONJUGACY IN A SPECIAL CLASS OF ARTIN GROUPS
Abstract (English):
The article gives an algorithm that allows solving the problem of weak power-law conjugacy of words in Artin groups, which are tree products of Artin groups with a tree structure and Artin groups of extra-large type.

Keywords:
Artin group, tree structure, weak power conjugacy, diagram

Let  – be a finitely generated Artin group with co-representation

where  – consisting of  alternating generators ,  – element of the matrix   :  [1].

The extra-large type of groups   was introduced by K. Appel and P. Schupp [1]. It is typical for it that  for all

For  we construct a graph  so that   correspond to the vertices of the graph , and – an edge with ends .  has a tree structure if  – is a tree.

which has a tree structure, is a free product with amalgamation by cyclic subgroups of Artin groups on two generators.

The woody structure for  in 2003 was identified by V.N. Bezverkhny [2].

Now

tree product  with tree structure or extra-large type, where the union ,  is taken over , , where  – is the generator of ,  – is the generator of .  will be called an Artin group with a generalized tree structure or a special class of Artin groups.

It is known that in Artin groups of extra-large type, with a tree structure, and in a special class of Artin groups, the problems of equality and conjugacy of words are solvable [1] - [3].

Let an annular map M, be given on the plane, that is, its complement - two components. We assume that K is unbounded, and H is the bounded component of M, while   is the outer, and   is the inner boundary of M. The cycle σ of the smallest length, including the edges , is the outer boundary cycle of M. Similarly, τ is the inner boundary cycle of M. Further, we denote a free product as F,  R symmetrized subset of F,   normal closure. If u and z are cyclically reduced in F, do not lie in N, are not conjugate in F, but conjugate in