On algorithmic problems in generalized tree structures of Coxeter groups

I. V. Dobrynina

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 477-490

Résumé

The main algorithmic problems in group theory are the problem of words, the problem of the conjugation of words for finitely presented groups, and the group's isomorphism problem. These problems were posed by M. Dehn.P. S. Novikov proved the unsolvability of the main algorithmic problems in the class of finitely presented groups. Therefore, algorithmic problems are studied in particular groups.Coxeter groups were introduced by H. S. M. Coxeter. A Coxeter group is a reflection group in which reflections with respect to hyperplanes limiting the fundamental polytope of the group are taken as generators. H. S. M. Coxeter listed all the reflection groups in three-dimensional Euclidean space and proved that they are all Coxeter groups and every finite Coxeter group is isomorphic to some reflection group in the three-dimensional Euclidean space which elements have a common fixed point.J. Tits studied Coxeter groups in the algebraic aspect. In his papers the problem of word in Coxeter groups is solved.It is known that in Coxeter groups the problem of the conjugacy of words are solvable and the problem of occurrence is unsolvable.K. Appel and P. Schupp defined a class of Coxeter groups of extra large type. Groups of this class are hyperbolic.V. N. Bezverkhnii introduced the notion of a Coxeter group with a tree structure. In a graph corresponding to a Coxeter group, one can always allocate the maximal subgraph corresponding to the Coxeter group with a tree structure. V. N. Bezverkhnii and O. V. inchenko solved a series of algorithmic problems in this class of groups.In the article the problems of the root and the power conjugacy of words in a generalized tree structures of Coxeter groups, which is a tree product of Coxeter groups of extra large type and of Coxeter groups with a tree structure. The proof of the main results uses the method of diagrams worked out by van Kampen, reopened by R. Lindon and refined by V. N. Bezverkhnii.

Détail

Keywords: Coxeter group, algorithmic problems, tree product of groups, diagram.