On the problem of conjugacy of words in a certain class of subgroups of Artin groups

V. N. Bezverkhnii ; N. B. Bezverkhnyaya

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 88-99

Résumé

One of the main problems in combinatorial group theory is the problem of equality and conjugacy of words. It is known that this problem is algorithmically unsolvable in the class of finitely defined groups. The problem arises of studying these problems in certain classes of groups, as well as whether subgroups of this class of groups inherit the algorithmic solvability of the word conjugacy problem. D. Collins and K. Miller defined a group with a solvable word conjugacy problem containing a subgroup of finite index in which the word conjugacy problem is not solvable. We also construct a group with an unsolvable word conjugacy problem containing a subgroup of finite index with a solvable word conjugacy problem. E. Artin defined braid groups and proved that the problem of word equality is algorithmically solvable in braid groups. A. A. Markov constructed an algebraic theory of braid groups and re-proved, using the constructed theory, the problem of word equality. F. Garside proved that the conjugacy problem of words in braid groups ${\mathfrak{B}}_{n+1}$ is solvable. Saito, using the ideas Of F. Garside, proved the solvability of the problem of equality and conjugacy of words in Artin groups of finite type. It is known that this class of groups belongs to braid groups. The interest is to study the solvability of this problem in subgroups of the class groups, in particular, in the normal divisor generated by the squares forming a group called painted subgroup of this group. In [1] it is proved that in a colored subgroup of Artin groups of finite type, the word conjugacy problem is solvable. It is known that in Artin groups with a tree structure, the word conjugacy problem is also solvable. [2]. In this paper, we prove that colored subgroups of Artin groups with a tree structure inherit the property of positive solvability of the word conjugacy problem.

Détail

Keywords: Artin groups, colored subgroup, word conjugation problem.