Involutions and Free Pairs of Bass Cyclic Units in Integral Group Rings

Let ZG be the integral group ring of the finite nonabelian group G over the ring of integers Z, and let ∗ be an involution of ZG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk,m(x), uk,m(x ∗)), or (uk,m(x), uk,m(y)), formed respectively by Bass cyclic and ∗-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ZG. 1. Bass Cyclic Units Let ZG be the integral group ring of the group G over the ring of integers Z, and let U(ZG) be its group of units. Suppose that ∗ is an involution of ZG that extends the involution ∗ of G. If u is a unit of ZG, then u∗ is also a unit, and we say that u is ∗-symmetric if u∗ = u. In this paper, we are interested in the nature of subgroups of U(ZG) of the form 〈u, u∗〉 or 〈v, w〉, where v and w are ∗-symmetric units. Indeed, if v and w are any two units, then we say that (v, w) is a free pair if 〈v, w〉 is a free group on the two generators. If B is a finite subgroup of G, we denote by B̂ ∈ ZG the sum of elements of B in ZG. Since (1− b)B̂ = B̂(1− b) = 0 for any b ∈ B, it follows that elements of the form (1 − b)aB̂, with a ∈ G, have square 0. The expression u = 1 + (1 − b)aB̂ is therefore a unit of ZG, and if B = 〈b〉 is cyclic, then u is called a bicyclic unit. It is clear that u 6= 1 if and only if a ∈ G does not normalize B = 〈b〉. In reference [3], it was shown that if G is a nonabelian finite group admitting an involution ∗, and if all Sylow subgroups of G are abelian, then ZG contains a free bicyclic pair (u, u∗). We can ask how general is this occurrence, and if it extends to other types of units. Recall that for large families of finite groups, see for example [5], the subgroup of U(ZG) generated by the bicyclic and Bass cyclic units has finite index in U(ZG). Thus we are led to ask if this latter result from [3] remains true if we replace bicyclic units by Bass cyclic units, as defined below. Let x be an element of G of order o(x) = d and write x̂ for 〈̂x〉. Then x x̂ = x̂x = x̂ for all integers j, and we define uk,m(x) = (1 + x+ · · ·+ xk−1)m + 1− k