**PHYSICS AND MATHEMATICS**

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**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