On problem of intersection of the adjacency classes of finitely generated subgroups of Coxeter's group with tree structure

O. V. Inchenko

Čebyševskij sbornik, Tome 17 (2016) no. 2, pp. 146-161

Résumé

P. S. Novikov in 1955–1956 showed the unsolvability of the main algorithmic problems in class of finite defined groups. In this connection there was important task of consideration of these problems in specific classes of finite defined groups. Thus, class of finite defined groups of Coxeter represents scientific interest. The class of groups of Coxeter was defined by H. S. M. Coxeter in 1934. The classe of finitely generated groups of Coxeter with tree structure was defined by V. N. Bezverkhnii in 2003. Let finitely generated group of Coxeter with tree structure is defined by presentation $$ G = \langle {a_1,...a_n ;(a_i )^2,(a_i a_j )^{m_{ij} }, i,j \in \overline {1,n}, i \ne j} \rangle $$ where $ m_{ij} $ — number which corresponds to a symmetric matrix of Coxeter. At that, if $i \ne j$, that $m_{ij} = m_{ji} $, $m_{ij} \ge 2$. If $m_{ij}=\infty$, that between $a_i$ and $a_j$ relation does not exist . The group matches finite coherent tree-graph $\Gamma$ such that: if tops of some edge -$e$ of graf Г are elements $a_i$ and $a_j$, that the edge $e$ matches relation $(a_i a_j )^{m_{ij}}=1$. On the other hand group $G$ may be represented as wood product of the two-generated groups of Coxeter, which are united by final cyclic subgroups. In this case, we will pass from graf Г of group $G$ to graf $\overline{\Gamma}$ as follows: we associate tops of some edge $\overline{e}$ of graf $\overline{\Gamma}$ groups of Coxeter with two generating elements $G_{ji}=\langle a_j, a_i; (a_j)^2,(a_i)^2, {(a_j a_i )}^{m_{ji}}\rangle$ and $G_{ik}=\langle a_i, a_k; (a_i)^2,(a_k)^2, (a_i a_k )^{m_{ik}}\rangle$, and edge $\overline{e}$ — cyclic subgroup $\langle a_i; (a_i)^2 \rangle$. The problem of intersection of the adjacency classes of finitely generated subgroups is that you need to find an algorithm that will help determine empty or not intersection $w_1H_1\cap w_2H_2$, where $H_1$ and $H_2$ any subgroup of group $G$ and $w_1, w_2\in G$. Previously, the author proved the solvability of this problem for free product with association of two Coxeter's groups with two generating element. In the article author shows solvability of a problem of intersection of the adjacency classes of finite number of finitely generated subgroups of Coxeter's group with tree structure. For this purpose group $G$ was presented as wood product of $n$ two-generated groups of Coxeter, which are united by finite cyclic subgroups. To prove of this result, the author used the method of special sets and method of types. These methods were defined V. N. Bezverkhnii. He used these methods for research of various algorithmic problems in free constructions of groups. Bibliography: 17 titles.

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Keywords: Coxeter's group with tree structure, problem of intersection of the adjacency classes, amalgamated free product, method of special sets, method of types.