ON FREE SUBGROUP IN ARTIN GROUP WITH TREE-STRUCTURE

Abstract

Let G be finitely generated Artin group with tree-structure defined by the presentation G =< a1, ..., an; < aiaj >mij=< ajai >mji , i, j = 1, n >, where mij is number that corresponds to symmetrical matrix of Coxeter, and mij > 2, i 6= j a group G matches the end coherent tree-graph Γ such that if the tops of some edge e of the graph Γ match the form ai and aj , then the edge e corresponds to the ratio of the species < aiaj >mij=< ajai >mji . Artin groups with a tree-structure was introduced by V. N. Bezverkhnii, theirs algorithmic problems were considered by V. N. Bezverkhnii and O. Y. Platonova (Karpova). The group G can be represented as the tree product 2-generated of the groups, united by a cyclic subgroups. We proceed from the graph Γ of the group G to the graph Γ the following follows: the tops of some edge e of the graph Γ put in correspondence Artin groups the two forming Gij =< ai , aj ; < aiaj >mij=< ajai >mji> and Gjk =< aj , ak; < ajak >mjk=< akaj >mkj>, and edge e will match cyclic subgroup < aj >. This paper considers the theorem on the freedom of the Artin groups with a tree-structure: let H be finitely generated subgroup of an Artin group G with a tree-structure, while for any g ∈ G and every subgroup Gij , i 6= j, executed equality gHg−1 ∩ Gij = E then H is free. In the proof of use of the ideas V. N. Bezverkhnii on bringing many forming of the subgroup to a special set.