ON NORMALIZERS IN SOME COXETER GROUPS

Abstract

Let G be a finitely generated Coxeter group with presentation G =< a1, . . . , an; (aiaj ) mij = 1, i, j = 1, n >, where mij — are the elements of the symmetric Coxeter matrix: ∀i, j ∈ 1, n, mii = 1, mij ≥ 2, i 6= j.

If mij ≥ 3(mij > 3), i 6= j, then G is a Coxeter group of large (extra-large) type. These groups introduced by K. Appel and P. Schupp.

If the group G corresponds to a finite tree-graph Γ such that if the vertices of some edge e of the graph Γ correspond to generators ai , aj , then the edge e corresponds to the ratio of the species (aiaj ) mij = 1, then G is a Coxeter group with a tree-structure.

Coxeter groups with a tree-structure introduced by V. N. Bezverkhnii, algorithmic problems in them was considered by V. N. Bezverkhnii and O. V. Inchenko.

The group G can be represented as tree product 2-generated of Coxeter groups, amalgamated by cyclic subgroups. Thus from the graph Γ of G will move to the graph Γ in the following way: the vertices of the graph Γ we will put in line Coxeter group on two generators Gij =< ai , aj ; a 2 i = a 2 j = 1,(aiaj ) mij = 1 > and Gjk =< aj , ak; a 2 j = a 2 k = 1,(ajak) mjk = 1 >, to every edge e joining the vertices corresponding to Gij and Gjk is a cyclic subgroup < aj ; a 2 j = 1 >.

In this paper we prove the following theorem: normalizer of finitely generated subgroup of Coxeter group with tree-structure G = Gij∗Gjk, Gij =< ai , aj ; a 2 i = a 2 j = 1,(aiaj ) mij = 1 >, Gjk =< aj , ak; a 2 j = a 2 k = 1,(ajak) mjk = 1 > finitely generated and exist algorithm for generating.