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

V. N. Bezverkhnii ; I. V. Dobrynina

Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 135-147

Résumé

The main algorithmic problems of group theory are the problems of words, conjugacy of words and the problem of isomorphism of groups. This algorithmic problems in the class of finitely presented groups are unsolvable. So the main algorithmic problems and their various generalizations are studied in certain classes of groups. Coxeter groups have been studied since 1934, and in the algebraic aspect — since 1962. The problems of words and conjugacy of words are algorithmically solvable in these groupss but the problem of occurrence is unsolvable. K. Appel and P. Schupp defined the class of Coxeter groups extra- large type in 1983. V. N. Bezverhny defined the Coxeter groups with a tree structure in 2003. The article discusses the generalized tree structures of Coxeter groups, which are the tree product of Coxeter groups of extra large type and Coxeter groups with a tree structure. The generalized tree structure of Coxeter groups, as well as the Coxeter group of extra large type, and a Coxeter group with a tree structure, refer to hyperbolic groups, so most of algorithmic problems algorithmically solvable, in particular, the problem of generalized conjugacy of words. The authors propose In this paper an original method for proving algorithmic solvable of the problem of generalized conjugacy of words in tree structures of Coxeter groups. This method uses G. S. Makanin's approach applied by Him to prove the finite generation of the normalizer of an element in braid groups. In addition, in this paper we show that the centralizer of a finitely generated subgroup in a generalized wood structure of Coxeter groups is finitely generated and there is an algorithm writing out its generators.

Détail

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