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Automorphisms of Direct Products of Groups and Their Geometric Realisations

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Automorphisms of Direct Products of Groups and Their Geometric Realisations Math. Ann. 263, 343 364 ~1983) Am Springer-Verlag 1983 Automorphisms of Direct Products of Groups and Their Geometric Realisations Francis E. A. Johnson Department of Mathematics, University College, Gower Street, London WC1E 6BT, UK O. Introduction An outstanding problem in the theory of Poincar6 Duality (=PD) groups is to decide whether each PD group can be realised as the fundamental group of a smooth closed aspherical manifold. See [2, 15-17] for previous work on this problem. There are two main ways of constructing new PD groups from old [18] ; (i) by extension of one PD group by another, and (ii) by torsion free extension of a PD group by a finite group. One approach to the above problem is to apply the constructions (i) and (ii) to known realisable PD groups and try to decide whether the resulting PD groups are realisable. Though other examples are known, for instance the examples con- structed by Mostow and Siu in [25], the most convenient PD groups with which to start are discrete uniform subgroups of noncompact Lie groups. If F is a torsion free discrete uniform subgroup of a connected noncompact Lie group G, and if K is a maximal compact subgroup of G, then F\G/K is a smooth closed manifold of type K(F, 1). The present paper is a continuation of both [15] and [16]. Whereas in [16] we allowed only Surface groups, that is, torsion free discrete uniform subgroups of PGLz(IR), and in [15] allowed only discrete uniform subgroups of Lie groups with no PSLz(IR) factors as building blocks, here we go some way towards combining the two. The bulk of the paper is a purely algebraic study of the automorphisms of products of certain groups, notably Surface groups. This may be thought of as a non-abelian variant of Wedderburn Theory. Its chief method is to introduce a quasi-ring structure into the set End(G) of endomorphisms of a group G, and to regard End(G (")) as a quasi-ring of n x n matrices over End(G), where G~") is the n-fold product of G with itself. Our main purely algebraic result [Corollary (2.2) below] may be stated simple as follows, where A S (7, denotes the wreath product A (") >d G~. 344 F.E.A. Johnson Theorem. Let G ~- G]m) x ... x G~p") be a direct product of groups such that (i) each G i has trivial centre and is not decomposable as a nontrivial direct product, and (ii) for i < j, there is no nontrivial homomorphism 0 : G i---, G j with O(G i) normal in G j. Then there is a natural isomorphism Aut(G)-~ fl aut(Gi)fO-p . i=1 This result is used in Sects. 4 and 5 to prove our main topological result, Theorem (5.4) below, which we state here thus: Theorem. Let I ~H x H'~G~Q--,1 be an exact sequence oJ groups where (i) H' is a finite product of Surface groups, (ii) H is a torsion free discrete uniform subgroup of a connected linear semisimple Lie group L with no compact factors, such that re(H) is non-discrete ./or any epimorphism rc from L to a Lie group of IR-rank 1, and (iii) there exists a smooth closed manifold of type K(Q, 1). Then there exists a smooth closed manifold of type K(G, 1). The link between the algebra and the realisation results is provided by the concept of weak rigidity. A PD group F is said to be weakly rigid if there exists a smooth closed manifold X of type K(F, 1) such that the natural map BDIFF(Xr)~BG(Xr) admits a right inverse up to homotopy, where DIFF(Xr) is the diffeomorphism group of X r, and G(Xr) is the monoid of homotopy equivalences ofX r. Our discussion of weak rigidity is carried out in Sect. 3. In the above theorem we allow the degenerate cases H~'= {1} and H = {1}. At the referee's suggestion, we first treat the case where H={1}. This is done in Sect. 4. We use this result to realise some 6-dimensional poly-Surface groups neglected in [16]. Finally, in Sect. 5 we treat the general case where H4= {1}. 1. Automorphisms of G (~) To distinguish between the notions of internal and external direct product, we will use the notation "G = A 1 ..... A k" to mean that the group G is the internal direct product of its normal subgroups A 1, ..., A k, reserving "A t x ... x Ak" for the external product, and "G =A~ ... Ak" merely to mean that G is generated by its normal subgroups A 1.... , A k. For any group G, G (") will denote the n-fold external direct product G(")=G x ... x G. L-- n --J Suppose that G=H ("). Then there is an obvious infective homomorphism :Aut(H)~O',--,Aut(H (")) defined by (~(a 1..... % or))(x 1.... , x,) = (al(x ~- 1(1))..... a,(x,-1(,j)), where, of course, Aut(H)IO', is just wreath product, i.e. semidirect product Aut(H) (") >~ 0",, where the action of the symmetric group 0", on Aut(H) (") is by ~(a 1.... , a.) =(a~_ ~(1~, ..., a,_ ~(,)). Automorphisms of Direct Products of Groups 345 Similarly, in the case where G=A 1 x ... x Ak, even with no repeated factors, there is an injective group homomorphism, still denoted by ~, :Aut(A1) x ... x Aut(Ak)~Aut(G ) given by (~(al, ..., ak)) (Xa, ..., Xk)= (al(xt), ..., a,(x,)). We shall show in a number of geometrically interesting cases, including products of Surface groups, that these natural maps : Aut(H)SO" ~Aut (H (")) : Aut(A1) x ... x Aut(Ak)~Aut(A 1 x ... x Ak) are isomorphisms. We may think of these as being semisimplicity results. The theory has a number of points of similarity with Wedderburn theory. The case where all factors are distinct is dealt with in Sect. 2. In this section we concern ourselves only with the case where all factors are the same, and we will prove Theorem 1.1. Let G be a nontrivial 9roup having the jollowing properties; (i) the centre, (~(G), oJ G is trivial, and (ii) G cannot be decomposed as a nontrivial direct product. Then, jor any neE+, the natural map : Aut (G)S 0", --+ Aut(G ~")) is an isomorphism. For any group G denote by End (G) the set of all group homomorphisms G~ G. End(G) is, of course, a monoid under composition and Aut(G) is its group of invertible elements. We recall briefly how to describe elements of End(G ~")) by means of matrices. This is commonplace when G is abelian, less so when G is nonabelian. First note that, in addition to its composition "product", End(G) also has a partially defined "sum" which we now describe. Definition. Let G--(G, *, 1) be a group and let ~, fl~ End(G). Say that ~ centralises fl iff Im~ centralises Imfl i.e. iff for all x, y e G, e(x)* [l(y)= fl(y), e(x). Define C(G)={(e, fl)eEnd(G)xEnd(G): ~ centralises fl}. Clearly C(G) is a symmetric subset of End(G)x End(G). The map A :C(G)--*End(G), (~Afl)(x)=e(x).fl(x), is our partially defined sum. In particular, denoting composition by juxtaposition of symbols, and abusing notation by writing "0" for the trivial homomorphism into the multiplicative group (G, *, 1), the following laws hold whenever all terms are defined. (0) ~1=1~--~, (I) (~Afl)y=(~7)A(fly), (II) 7(aAfl)=(7~)A(Tfl), (III) ~A(flAy)=(eAfl)AT, (IV) eAfl=flAe, (V) eA0=0Ae=e. 346 F.E.A.Johnson DeJinition. By a quasi-ring we mean a sextuple (A, C, A,., 1,0) where A is a set, C C A x A is a symmetric subset, 0, leA, and A : C--, A, : A x A ~ A are mappings satisfying (i) for each a,b, ceA, (a,b)sC, and (aAb, c)sC if and only if (b,c)EC and (a, bdc)s C, (ii) for each a, b, csA, (a, b)e C iff (ca, cb)e C iff (ac, bc)s C, (iii) for each as A, (a, 0)e C, and (iv) the laws (0)-(V) above are satisfied when all terms are defined Clearly (End(G), C(G), A,., 1, 0) is a quasi-ring for any group G. Definition. Define 92 - e=(e~j)~ <~=<" eiY End(G) , the set of all n x n matrices in ~ <j<. / End(G) for a group G, and put M,~ = { s 9~: for each i,j, ks { 1, .. ., n}, c~ij centralises c~ik provided j 4= k} Now let ~,flsM~. Then for each r,s and each i,j with i+j, observe that c%ifli~ centralises e~ifljs. Thus we may define a map [] . M.G x M,,G -~ 92,G by i.e. for all xs G (~lSfl)r~(x) = ~rl(flls(X))*"" *~,.(fl.s(X)), where "*" is multiplication in G. Of course, the above is by straightforward analogy with ordinary matrix multiplication over a genuine ring Proposition 1.2. a[-lfle My for all c~, fl E M,.G Proofi We must show that (a[~fl)rs centralises (c~Rfl)rt for s + t. For x, ye G we have (~ *g2 * .'- *g,*hl *h2* "" .h,, where gi=c%i(fli~(x)) and hj=erj(flj,(y)). Since ctri centralises c~j for i#j, then gi*hj = hj*g i for i#j, and we may, rearrange the above product to get (~V-lfl)~(x).(cd-]fl),t(y) =gl *hi *g2* h2 * "'" *g,*h, However, gi* hi = ~ flit(Y)) = ~ri(flu(Y) * fli~(x)) since fli~ centralises flit for s + t.
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