On subgroups in Artin groups with a tree structure

Abstract

In the article, the author continues to consider issues related to the problem of freedom in Artin groups with a woody structure, and published jointly with V. N. Bezverkhnim in the Chebyshev Collection in 2014. In particular, the following subgroup theorem is proved for Artin groups with a tree structure: if 𝐻 is a finitely generated subgroup of the Artin group with a tree structure, and the intersection of 𝐻 with any subgroup conjugate to a cyclic subgroup.
generated by the generating element of the group, there is a unit subgroup, then there is an algorithm describing the process of constructing free subgroups in 𝐻.
The study of free subgroups in various classes of groups was carried out by many outstanding mathematicians, the fundamental results are presented in a number of textbooks on group theory, monographs and articles.
Artin’s groups have been actively studied since the beginning of the last century. If the Artin group corresponds to a finite tree graph such that its vertices correspond to generating groups, and every edge connecting the vertices corresponds to a defining relation connecting
the corresponding generators, then we have an Artin group with a tree structure.
An Artin group with a woody structure can be represented as a tree product of twogenerators Artin groups united by infinite cyclic subgroups.
In the process of proving the main result, the following methods were used: the reduction of the set of generators to a special set introduced by V. N. Bezverkhnim as a generalization of the Nielsen set to amalgamated products of groups, as well as the representation of a subgroup as a free product of groups and the assignment of a group using a graph.