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On the Whitehead groups of semi-direct products of free groups Choo, Koo-Guan

Abstract

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 K₀Z(G). Then Wh G = 0 if G is free abelian (Bass-Heller-Swan), free (Gersten-Stailings) or a semi-direct product of a free group and an infinite cyclic group (Farrell-Hsiang) and K₀Z(G) = 0 if G is free abelian (Bass-Heller-Swan), free (Bass) or a direct product of a free abelian group and a free group (Gersten). In this thesis, we extend these results to a wider class of groups. Let α be an automorphism of G and F a free group. We denote the semi-direct product of G and F with respect to a by- G x[sub α] F. New, let D be a direct product of n free groups and α an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed. First, by using techniques of Bass-Heller-Swan on Whitehead groups of certain direct products, together with techniques of Stallings on Whitehead groups of free products, we prove Wh D = 0 and K₀Z(D) = 0. Next, we establish a fundamental theorem for coherent rings : If R is a (right) Noetherian ring and if G₁ and G₂ are groups such that the group rings R(G₁) and R(G₂) are (right) coherent, then R(G₁ * G₂) is (right) coherent, where G₁ * G₂ is the free product of G₁ and G₂. A similar theorem has been anounced by Fi Waldhausen. From this fundamental theorem, we deduce that if A is a free abelian group and F is a free group, the integral group ring Z(A x F) of A x F is (right) coherent. If A is of finite rank, then Z(A x F) has finite right global dimension. Combining these facts with techniques of Farrell-Hsiang on Whitehead groups of certain semi-direct products of groups and using the triviality of Wh D and K₀Z(D), we show that Wh(D x[sub α] T) = 0 and K₀Z(D x [sub α] T) = 0. The first result generalizes that of Farrell-Hsiang on semi-direct product F x[sub α] T, and the second result implies, in particular, that for the fundamental group π₁(M) of a closed surface M (other than the real projective plane), the projective class group of Z(π₁ (M)) is trivial. If M is a closed surface (other than the real projective plane) and (S¹)[sup k] is the k-dimensional torus, the fundamental group of M x (S¹)[sup k] Then the triviality of Wh(D x[sub α] T) implies the following result in topology : If N is a differentiable or PL manifold of dim ≥ 5 which is h-cobordant to M x (S¹)[sup k], then N is actually i diffeomorphic or PL-homeomorphic to M x (S¹)[sup k] respectively. Finally, by adapting Gersten's discussion on Whitehead group of free associative algebra to the case of a twisted free associative algebra, and by using the facts that Wh(D x[sub α] T) =0 and K₀Z(D x[sub α] T) = 0, we prove Wh((D x[sub α] T) x [sub αxid[sub T]] F) = 0. The factor T can presumably be dropped, although this is not entirely obvious. There is also a separate chapter on combinatorial group theory in which we give certain necessary and sufficient conditions for a given one relator group to be of the form F x [sub α] T.

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