On problem of generalized conjugation of words in a generalized tree structures of Artin groups

A. S. Ugarov   ; I. V. Dobrynina

Čebyševskij sbornik, Tome 25 (2024) no. 3, pp. 236-247

Résumé

Artin groups are a generalization of known braid groups, in which the problems of words and conjugacy of words are algorithmically solvable. Due to the complexity of solving these problems in the Artin group class, algorithmic problems are considered in its various subclasses. In 1983 K. Appel and P. Schupp defined the Artin groups extra-large type. In 2003, V. N. Bezverkhny introduced the Artin group with a tree structure. Artin groups of extra-large type and Artin groups with tree structure are well studied and most of the algorithmic problems are solved in them, in particular, the algorithmic solvability of the problem of generalized conjugacy of words is proved. The article deals with generalized tree structures of Artin groups, which are tree products of Artin groups of extra-large type and Artin groups with a tree structure, united by cyclic subgroups corresponding to generatings these groups. The authors provide a original proof of algorithmic solvability of the problem of generalized conjugacy of words in generalized tree structures of Artin groups. The method of proof uses the approach of G. S. Makanin, applied by him to study the finite generality of the element Normalizer in braid groups. In addition, this paper shows that the centralizer of a finitely generated subgroup in the generalized tree structure of Artin groups is finitely generated and there is an algorithm that writes out its generators.

Détail

Keywords: algorithmic problems, Artin group, generalized conjugation, tree product of groups, centralizer.