Whitehead groups of certain semidirect products of free groups

Abstract

Let D=F1 x F2 x... x Fn be a direct product of n free groups F1, F2, * , F* * , ox an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed, T an infinite cyclic group and F another free group. Let D x a T be the semidirect product of D and T with respect to a and (D x a T) x aXIdT F the semidirect product of D xa Tand F with respect to the automorphism x id T of D Xa T induced by a. We prove that the Whitehead group of (D xa, T) X 2xidT F and the projective class group of the integral group ring Z((D x a T) X aXidT F) are trivial. These results extend that of [3]. Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring Z(G) of G by kOZ(G). We recall the definition of semidirect product of groups and the definition of twisted group ring. For undefined terminologies used in the paper, we refer to [3] and [4]. Let oc be an automorphism of G and F a free group generated by {tA}. If w is a word in tA defining an element in F, we denote by Iwl the total exponent sum of the tA appearing in w. The semidirect product G xa F of G and F with respect to a is defined as follows: G x . F=GxF as sets and multiplication in G x . Fis given by (g, w)(g', w') = (go-lwl(g'), ww'), for any (g, w), (g', w') in G x F. In particular, if F is an infinite cyclic group T= (t) generated by t, we have the semidirect product G x a T of G and T with respect to oc. Let R be an associative ring with identity and oc an automorphism of R. Let F be a free group (or free semigroup) generated by {tA}. The otwisted group ring R,[F] of F over R is defined as follows: additively R,[F]=R[F], the group ring of F over R, so that its elements are finite linear combinations of elements in F with coefficients in R. Multiplication in R,[F] is given by (rw)(rIw')=roc-1I1(r')ww', for any rw, r'w' in R,[F]. In particular, if F is a free group (resp. free semigroup) generated by t, we Received by the editors May 25, 1973. AMS (MOS) subject classfiJcations (1970). Primary 13D15, 16A26, 18F25; Secondary 16A06, 16A54.