On subgroups in Artin groups with a tree structure

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 two-generators 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.