On normalizers in some Coxeter groups

I. V. Dobrynina

Čebyševskij sbornik, Tome 17 (2016) no. 2, pp. 113-127.

Résumé

Let $G$ be a finitely generated Coxeter group with presentation $$G= a_1,\ldots, a_n;(a_ia_j)^{m_{ij}}=1, \, i,j =\overline{1,n} >,$$ where $m_{ij}$ — are the elements of the symmetric Coxeter matrix: $\forall i,j \in\overline{1,n},\, m_{ii}=1,\,m_{ij} \geq$ $ \geq2, \, i\ne j$. If $m_{ij}\geq3$ $(m_{ij}>3)$, $i\ne j$, 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 $\Gamma$ such that if the vertices of some edge $e$ of the graph $\Gamma$ correspond to generators $a_i, a_j$, then the edge $e$ corresponds to the ratio of the species $(a_ia_j)^{m_{ij}}=1$, 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 $\Gamma$ of $G$ will move to the graph $\overline{\Gamma}$ in the following way: the vertices of the graph $\overline{\Gamma}$ we will put in line Coxeter group on two generators $$G_{ij} = , a_j; a_i^2=a_j^2=1,(a_ia_j)^{m_{ij}}=1>$$ and $$G_{jk} = , a_k; a_j^2=a_k^2=1,(a_ja_k)^{m_{jk}}=1>,$$ to every edge $\overline{e}$ joining the vertices corresponding to $G_{ij}$ and $G_{jk}$ is a cyclic subgroup $$;a_j^2=1>.$$ In this paper we prove the following theorem: normalizer of finitely generated subgroup of Coxeter group with tree-structure $$\overline{G}=G_{ij}\ast_{; \ a_j^2>}G_{jk},$$ $$G_{ij} = , a_j; a_i^2=a_j^2=1,(a_ia_j)^{m_{ij}}=1>,$$ $$G_{jk} = , a_k; a_j^2=a_k^2=1,(a_ja_k)^{m_{jk}}=1>$$ finitely generated and exist algorithm for generating. Bibliography: 18 titles.

Détail

Keywords: Coxeter group, tree-structure, normalizer, amalgamated product.

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