On free subgroups in generalized tetraedron groups of type (2, 2, <i>n</i> , 2, 2, <i>m</i> )
The Tits alternative for generalized tetrahedron groups is investigated in the paper. A group G satisfies the Tits alternative, if it contains either a solvable subgroup of finite index or a free group of rank 2. Generalized tetrahedron groups have a presentation of the form Γ = x1, x2, x3|x1k1 = x2k2 = x3k3 = R12(x1, x2)m12 = R23(x2, x3)m23 = R13(x1, x3)m13 = 1>, where k1, k2, k3, mij ≥ 2; Rij(xi, xj) is a cyclically reduced word in the free product xi|xiki = 1> ∗ xj|xjkj = 1>, which is not a proper power. There is a conjecture that these groups satisfies the Tits alternative. At present this conjecture is proved not for all types of generalized tetrahedron groups. In this paper we consider a class of groups of a type Γ = a, b, c|a2 = b2 = cn = (ab)2 = (acα)2 = Rm (b, c) = 1>, где m ≥ 3; n ≥ 6.. The following theorem is proved. Theorem. Let Γ = a, b, c|a2 = b2 = cn = (ab)2 = (acα)2 = Rm (b, c) = 1>, где m ≥ 3; n ≥ 6. Than if Γ is different from the group Γ1 = a, b, c|a2 = b2 = c6 = (ab)2 = (ac)2 = (bc)3 = 1>, then Γ contains a non-abelian free subgroup. The group Γ1 is almost solvable.