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.
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 : .
The extra-large type of groups was introduced by K. Appel and P. Schupp . 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 .
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  - .
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