ОБ АЛГОРИТМЕ РЕШЕНИЯ ПРОБЛЕМЫ СОПРЯЖЕННОГО ВХОЖДЕНИЯ В ЦИКЛИЧЕСКУЮ ПОДГРУППУ В НЕКОТОРЫХ ДРЕВЕСНЫХ КОНСТРУКЦИЯХ ГРУПП АРТИНА

Abstract

P.S. Novikov established the unsolvability of the problems of words, conjugacy of words in the general class of groups given by a finite set of generating words and defining words. These problems, as well as their various generalizations, have become the subject of study in specific groups. In particular, this article discusses one of such problems in the construction of groups, which, on the one hand, is a tree product, and, on the other hand, refers to Artin groups. We prove the solvability of a conjugate occurrence in a cyclic subgroup in a given class of groups. At the same time, not all Artin groups are taken, but only those in which, in determining ratios, the number of alternating generators on each side is more than three, or those that themselves have a tree structure. The cyclic subgroups generated by the degrees of the corresponding generators are taken as the combined subgroups. The groups introduced by Artin are quite complex, the study of algorithmic problems in them presents difficulties and general results have not yet been published. However, a large number of special cases of these groups with algorithmic issues studied in them are considered. These groups have been of scientific interest since the beginning of the last century, generalizing the well-known braid groups.