On an algorithm for solving the problem of power conjugacy of words in generalized tree structures of Artin groups

Abstract

The article considers the solution of the problem of power conjugacy of words in generalized tree structures of Artin groups by geometric methods based on the study of diagrams over this class of groups having a single-layer structure, as previously shown by the authors. Chart transformations are used, including abbreviations introduced by M. Den and V. N. Bezverkhnim.
The article is a continuation of the consideration of algorithms for solving problems of combinatorial group theory in generalized tree structures of Artin groups, previously the authors proposed algorithms based on a diagram approach to solve conjugacy problems, generalized conjugacy of words, the construction of centralizers of an element and a finitely generated subgroup.
The class of groups considered in the article is a tree product of Artin groups with a tree structure and Artin groups of extra-large type, amalgamated by cyclic subgroups corresponding to the generators of the groups.
Artin groups were introduced at the beginning of the last century as a generalization of the well-known braid groups, the class of extra-large Artin groups was isolated in 1983, the class of Artin groups with a woody structure was isolated in 2003. The groups considered in this paper belong to almost large Artin groups and the problems of words and conjugacy of words are algorithmically solvable in them, which follows from the proof of their biautomaticity. The
approach proposed by the authors in solving the problem of power conjugacy of words is more
visual and simple.