The problem of the conjugation of words in $HNN$-extension with a finite number entrance letters of a tree product of cyclic groups with cyclic amalgamation

E. S. Logacheva

Tula State Pedagogical University

Abstract: In the work of the positive solution of the conjugation of words in $HNN$-extension with the system of entrance letters. The base $HNN$-extensions is a wood product of the infinite cyclic groups with cyclic subgroups. The result is a generalization of the conjugacy problem in $HNN$-extension of a wood product of cyclic groups associated cyclic subgroups with one entrance letter.
The conjugacy problem for words is of interest in free designs groups. The problem was solved in free groups with cyclic subgroups by S. Lipshutz, in the $HNN$-extension of a free group by an associate of cyclic subgroups by A. Friedman, in $HNN$-extension of a tree product with the association cyclic groups associated with cyclic subgroups by author with V. N. Bezverkhny.
In this paper a positive solution of the conjugation problem for words in $HNN$-extension with the system of entrance letters. The base $HNN$-extensions is a tree product of the infinite cyclic groups with cyclic subgroups. The result is a generalization of the conjugacy problem in $HNN$-extension of a wood product of cyclic groups associated cyclic subgroups with one entrance letters. Assertion is proved for any number of entrance letters using the method of mathematical induction. In the proof of the main theorem the author proved self result assertion :

• algorithmic solvability of intersection of finitely generated subgroup of the core group with an associated sub-group;
• algorithmic solvability of intersection of the related class of finitely generated subgroup of the core group with an associated sub-group.

Bibliography: 13 titles.

Keywords: the group, the subgroup, the $HNN$-extension, the tree product, the conjugacy problem.

UDC: 519.4

Received: 24.04.2014