On the Tits’ alternative for subgroups of $F$-groups

V. G. Durnev ; O. V. Zetkina ; A. I. Zetkina

Čebyševskij sbornik, Tome 15 (2014) no. 1, pp. 110-120.

Résumé

Tits proved that for any finitely generated linear group $G$, the following statement holds: $G$ is either solvable-by-finite, or it contains a subgroup isomorphic to the free group $F_2$ of rank $2$. This leads to the concept of the Tits' alternative for a class of groups: For a class $C$ of groups the Tits' alternative holds, if an arbitrary group $G$ from this class is either solvable-by-finite, or it contains a subgroup isomorphic to the free group $F_2$ of rank $2$. A number of works have addressed the studying of the classes of groups for which the Tits' alternative holds. The Tits' alternative is related to the following problem which has been independently studying for a long time in combinatorial group theory: Find the class of groups possessing the following property: for an arbitrary group $G$ from this class, the following alternative holds: either a non-trivial identity holds on the group $G$, or $G$ contains a subgroup isomorphic to the free group $F_2$ of rank $2$. For subgroups of the groups with one defining relation, this problem was fully studied by D. I. Moldavanskii, A. A. Chebotar', A. Karrass and D. Solitar. For groups satisfying small cancellation conditions, this problem was studied by V. P. Klassen in describing the subgroups of such groups. The full description of Abelian subgroups of arbitrary $F$-groups is given in the famous monograph by R. Lindon and P. Schupp. In the present work, this result is strengthened: we give a description of subgroups of $F$-groups, on which a non-trivial identity holds and prove the Tits alternative for subgroups of $F$-groups. More accurately, we prove that for the subgroups of Fuchsian groups, the strengthened variant of the Tits' alternative holds: An arbitrary subgroup $H$ of a Fuchsian group either is solvable group of degree $\leq 3$ or alternating group $A(5)$, or $H$ contains a subgroup isomorphic to the free group of rank $2$, No non-trivial identity does hold on a subgroup $H$ of an arbitrary Fuchsian group $G$ if and only if $H$ contains a subgroup isomorphic to the free group $F_2$ of rank $2$.

Détail

Keywords: Fuchsian groups, $F$-groups, Tits' alternative, group satisfying a non-trivial identity.

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