Some Residually Finite Groups Satisfying Laws

We give an example of a residually-p finitely generated group, that satisfies a non-trivial group law, but is not virtually solvable. Denote by Fn the free group on n generators. Recall that, given a group word m(x1, . . . , xn) ∈ Fn, a group G satisfies the law m = 1 if for every u1, . . . , un ∈ G, m(u1, . . . , un) = 1. Given a set S of group laws, the n-generator free group in the variety generated by S is the quotient of Fn by the intersection of all kernels of morphisms of Fn to a group satisfying all the group laws in S . Taking the quotient by the intersection of all finite index subgroup (resp. of p-power index), we obtain the restricted (resp. p-restricted) n-generator free group in the variety generated by S . The celebrated Tits Alternative states that if G is a finitely generated linear group over any field, then either G contains a non-abelian free subgroup, or it is virtually solvable (i.e. contains a solvable subgroup of finite index). It follows that if such a group G satisfies a nontrivial group law, it is virtually solvable. It is natural to ask to what extent the assumption of linearity can be relaxed. Can we, for instance, replace linearity by residual finiteness? Here we show that this is not possible, even under the assumption that G is residually-p (i.e. residually a finite p-group). We provide several constructions. The results we obtain are probably known to the specialists; however, to the best of our knowledge, they do not seem to appear in the literature. For any q ∈ N, let Gq be the restricted free 2-generator group in the variety generated by the group law [x, y] = 1. We begin by the following elementary result: Theorem 1. For q = 30, Gq is a 2-generator, residually finite group that satisfies a nontrivial group law, but is not virtually solvable. Proof. The only nontrivial verification is that G30 is not virtually solvable. To show this, it suffices to show that, for every n, G30 has a finite quotient having no solvable subgroup of index ≤ n. Start with I = Alt5. Then |I| = 60 and any solvable subgroup of I has order ≤ 12. Therefore, for every m, any solvable subgroup of I has order ≤ 12, therefore index ≥ 5. Now, for all k ≥ 2, the wreath product I ≀Ck = I k ⋊Ck is generated by 2 elements: the element (1, z), where z is a generator of Ck (we now identify z and (1, z)), and the element σ = ((s, t, 1, . . . , 1), 1), where s has order 2, t has order 3, and s and t generate I. Indeed, σ = (s, 1, . . . , 1), zσz = (t, 1 . . . , 1), so that σ and z generate I ≀ Ck. Now I ≀ Ck has derived subgroup of exponent 30, hence it is a quotient of G30. Date: April 24, 2006.