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 ~ [] . 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. Thus gi* hi = hi* gi, and we may rearrange the above product again to get (~[--]fl)rs(x)*(al--'lfl)r,(Y)=ht *gl *h2*g2* "" *h.*gn =hi *h2 * "'" *h.*gl *g2* "" *gn =(at-lfl),,(y)*(~[~fl),~(X). Q.E.D. Automorphisms of Direct Products of Groups 347 Thus [] is actually a map []'M, ~ x M,G ~M,.G Moreover, defining 1,eM~, by (I")~={~ do if rq=sr=S then I, is a two-sided identity for (M,a, UI). Proposition 1.3. M .e = ( M ~. , [], 1.) is a monoid, for any group G. Proof It remains only to show that [] is associative, the proof of which follows the usual proof that matrix multiplication is associative. The only properties that we need of (End(G), A) are (I)-(IV) mentioned above. Q.E.D. For any group G, let n r : G~")-+ G be the r th projection, n,(x~ ..... x,) = x r, and let is:G--+G ~") be the canonical inclusion of the s th factor, is(x)=(1, 1.... ,x, ..., 1). We define a mapping s 'h place cb:End(GC"))--+Td~ by q~(~)rs=nr~is. Proposition 1.4. For any group G, cb" End(G~"))-+M,~ is a monoid isomorphism. Proof. Observe that c~is centralises ei t for s =t= t, so that rtr~i~ centralises n,ei, for s =4=t. Hence Im(~b) is certainly contained in M,a. To see that cb is a monoid homornor- phism, first note that each mapping q~r~ : End(GC"))--+End(G), q~,s(~)= n~c~i~, is A-homomorphism If l eEnd(G ~")) is the identity map we have 1 =(ilZta)d(iz:gz)A ... A(i.~,), so that o~fl = (o:it rq fl)A(o~izztzfl)A ... A(o~i,,rc,,fl) , for all e, fleEnd(Gt")). And since ~0 is a A-homomorphism we get q0~(0cfl) = ~0~(c~iI rq fl)A ... A qO~s(~i,~,fl) But q~,~(o~ipz~pfl)= q~ p(e)q~p~(fl). Hence = (~(c~) [] 4,(/~)),~. That is, 4~ is a monoid homomorphism. Injectivity of q~ is straightforward, and we leave it as an exercise. Surjectivity, also straightforward, is more instructive. Let (e~s)l _<~_<,eM, a, so that ~,g centralises l~=s~n %q for p*q. For each r define e~:Gt")-+G by c~(x I .... ,x,)=cc,~(xx)*... *~,(x,). Then e, is a homomorphism by the centralising hypothesis, and we get a homomorphism e : G t")--+ G (") by c~(x)= (e~(x) ..... c~,(x)). Moreover ~(e),, = r so that is surjective. Q.E.D. In the proof of Theorem (1.1), we shall need Lemma 1.5. Let G be a nontrivial group having the .[ollowing properties; (i) G has trivial centre, (ii) ,G is not expressible as a non-trivial direct product, and 348 F.E.A.Johnson (iii) G= A 1 ... A, where each A i is normal in G, and A i centralises Aj for i:~j. Then/or some unique re{1 ..... n} we have G=A,, and A~={1}, for s+r. Proof. By induction on n. For n-- 1, there is nothing to prove. Suppose proved for n- 1, and put B=A~ ... A,_ ~. Then B is normal and centralises A,, so that Bc~A, centralises G=B.A,. But G has trivial centre, so that Bc~A~= {1}, and G~-B x A,. Since G is not isomorphic to a non-trivial direct product, we get either that B = { 1} or A, = { 1 }. If A, = { 1 }, then G = A~ ... A,_ ~, and we get the result by induction. If B={1}, then G=A,, and the result follows. Q.E.D. Consider the mapping ~u : Aut(G)~,) x o-~M, G defined by ~g(e,a), = {; ~ r=~(s), r#:tr(s), where ~=(c~i)1_ It is straightforward to verify that 7j induces an injective group homomorphism ~u: Aut(G)S0- --,(M,) * , where (M,G) * denotes the group of invertible elements of the monoid M,G. Proof of Theorem 1.1. Let G be a nontrivial group with trivial centre, and suppose that G is not expressible as a nontrivial direct product. Let cte Aut(G (")) and write ~(~) = (c%) 1 <, <,. Fix ie { 1..... n}. Now ni(G ~"~)= G so that G = Im(ail) Im(ai2) ... 1-----S~n Im(c%). Since Im(cqs) centralises Im(c%) for s 4: t, then by Lemma (1.5), there exists a unique tr(i)e {1 ..... n} such that Im(cti~(1))= G, with Im(cto)= {1} if i:t=j. We claim that the mapping tr:{1 ..... n}--*{1 .... ,n} is bijective. For if not, then cr is not surjective, and there exists me {1, ..., n} such that %,=0 for all r. Thus G~Im(im)CKer0z,c0 for all r, so that G is trivial. Contradiction. Hence tr is bijective. Also note that each c~potp~ must be an automorphism of G, for if ep~(p) is not surjective neither is ~zp~, which is a contradiction, whilst if O~ptr(p) is not infective neither is O~ip, also a contradiction. Thus Ctp~(p)eAut(G) and we have shown that ~(Aut(G ("~) = ~(Aut(G S 0",). Since ~b maps Aut (Gt")) bijectively onto (M,)G* , we see that - 1 7j : Aut(G)~o. ~Aut(Gt.)) is a group isomorphism. It is easy to see that 4- x 7j is in fact the previously defined map ~. Q.E.D. [Theorem (1.1)] By a Surface group we mean the fundamental group of a closed connected 2-manifold of genus >2. The hypotheses imposed on G in (1.1) are satisfied by Surface groups and by many others. The following list is representative, without in any way attempting to be exhaustive. Proposition 1.6. Any group G from the classes ~1,~2,~ 3 below satisfies the hypotheses of (1.1), namely that G has trivial centre and is not decomposable as a nontrivial direct product. ~1 : Surface groups ~2 : Nontrivial free products. In particular, free groups of rank > 2. if'3: Discrete subgroups o/finite covolume in simple noncompact Lie groups with trivial centre. Automorphisms of Direct Products of Groups 349 Pro@ The proof for E 3 is a by-now-standard argument using Borel's Density Theorem, and is almost, but not quite, proved in Chap. V of Raghunathan's book [26]. If G=GloG 2 is discrete with finite covolume in LCGL(n,~,)EGL(n,I~), where L is a simple noncompact Lie group with ~(L)= 1, put L~---(Zariski closure of GI)c~L. Then Gi 2. Product Structures: Uniqueness and Automorphisms Let E be a class of groups. By a E-product structure on a group G we mean a finite sequence ~ = (Ai, Gi, Pi) l -integer such that (iv) G~ ~ Gj if i 4:j (v) G=A 1 ..... A, and (vi) A i ~- G--i ~v'~" There are a number of possible notions of uniqueness for E-product structures. We consider two. Definition. If ~ = (Ai, Gi, Pi) 1 <-i <-, and `3 = (B i, Hi, qi) 1 <__i <__m are E-product structures on a group G say that ~ and .3 are equivalent if and only if n = m and for some (necessarily unique) ae 0", we have H i-___G~,) and ql = p~,). Moreover, we say that and `3 are strongly equivalent when they are equivalent and in addition B~ = A~), for all i, 1 < i < n. Note that we take "equals" here, not merely "~". In this section we give conditions on a class of groups E which ensure that when G has a E-product structure, it is unique in the strong sense. Thus, without further mention, for the remainder of this section, E will denote a class of nontrivial groups satisfying the following three properties; (~31) Each GeE has trivial centre (~ILI2) If GeE then G is not decomposable as a nontrivial direct product (~3) There is a total ordering < on the isomorphism types of E such that, if H < K then there is no nontrivial homomorphism 0 :H-*K with O(H) normal in K. Here by H < K we mean that H < K and H ~ K. In this section we shall prove Theorem 2.1. If 6, ~ are E-product structures on a group G, then ~ and `3 are strongly equivalent. 350 F.E.A.Johnson Before embarking on the proof of (2.1), we give some explanations and examples. Observe that in order to get any sort of uniqueness result it is necessary to exclude, as we have done, the trivial group from the class E. However it is useful to be able to refer to the trivial group, which we do by putting E+ =(s group} and extending < to a total ordering on E+ by insisting that H < { 1 } for all He E. Then the statement "H < K" will mean at least "H + {1 }". The simplest example of a class satisfying (~1), (~32), (~33) is the class of finitely generated free groups of rank > 2. In this case we take H < K iff rank(H) < rank(K). The class which interests us most is the class of Surface groups, though here the ordering is slightly more complicated, and we refer the reader to Proposition (2.7), where it is explained in detail. Theorem (2.1) has the following obvious corollary, which is, in fact, the main purpose of this section. Corollary 2.2. Let (A i, G i, Pi)~ <_is, be a ~-product structure on G. Then the natural maps ~ defined in Sect. 1 give isomorphisms Aut(G)~ fi Aut(Ai)~ [~ Aut(Gi)~O'p . i=1 i=1 On the way to proving (2.1), we first prove Proposition 2.3. Ij A, BeE and At")_~B Proo~ Let /:A~n~--~B~m) be an isomorphism, and let ~r:Br and is:A~A ~"~ denote the canonical projections and inclusions. Put Brs = Im(~zrji~). Then B= Brl .Br2 ... Brn and B,p centralises Brq for p#q. By Lemma (1.5), there is a function r : { 1..... m} ~ { 1..... n} with the property that, for all re { 1..... m}, B = B,.,r whilst B~s=l for s#cr(r). Note that nrJi~r is onto, so that B Lemma 2.4. Suppose that G= A oB, where A is a centreless characteristic subgroup of G. Then B is also characteristic in G, and : Aut(A) x Aut(B)~ Aut(G) is an isomorphism. Proof Let Aut(~) be the group of all automorphisms of the exact sequence @=(I~A ~G P-~B~I). Since (~ is a trivial extension, there is a split exact sequence 1 ~ C(~)~ Aut(i~)~Aut(A) x Aut (B)--* 1 Automorphisms of Direct Products of Groups 351 where 0 is the obvious restriction map, the splitting map ~ is the one defined in Sect. 1, and C(~) is the group of congruences of ~, i.e. the group of automorphisms which induce the identity on A and B, the "ends" of (~. However, since A has trivial centre, C(~) is trivial. For if he C(~), then for each XE G, p(h(x)x- i) = ph(x)p(x)- x = p(x)p(x)-I = 1 hence h(x)x- 1 e Ker(p) = A. Consider the map/~ :G--+A defined by/~(x) = h(x)x- 1. If a~A and x~G then f~(xa)=f~(x), since f~(xa) = h(x)h(a)a- ix- 1 = f~(x), h inducing the identity on A. Now if x~G, then for all yeA, yf~(x)y- 1 = h(y)h(x)x- l y- t = f~(xy), and, by the above, /~(yx)= [z(x(x-lyx))=fl(x), since x-lyxeA. That is, yfJ(x)y-1 =/~(x) for all yeA, so that each/l(x)e3(A), the centre of A. Since 3(A)= {1}, then for all xe G, h(x)= x. Thus C((~)= {1}. Now Aut((~)= Aut(G), since A is character- istic in G, so that ~ is an isomorphism ~ :Aut(A) x Aut(B)~Aut(G) with inverse r and B is characteristic since it is visibly invariant under the whole of Aut(G)~Aut(A)x Aut(B). Q.E.D. Lemma 2.5. Suppose that G = A oB = A' oB' and A ~ Gtxp) (p > 1), B _~ (] G i, A' ~- H]q~ i=2 (q>l), B '~- [] Hj where Gi, Hje~ +, possibly allowing repetitions amongst the j=2 Gi, H j, and such that G 1 Proof First note that everything will follow if we can prove that A C A'. For then it follows by symmetry that A = A'. The conclusion that G 1 ~ H 1 and p =q will follow from Proposition (2.3). Also note that A is then characteristic, for if 0r is an automorphism of G, putting A"= e(A) and B"= e(B) we see that the decomposition G=A"oB" satisfies our hypotheses, so that A=A"=e(A). Since a product of centreless groups is centreless, A has trivial centre so that B is also characteristic by (2.4). To show that B=B', note that B~-G/A=G/A'~-B'. IfJ:B~B' is an isomorphism consider the automorphism h:G~G defined by h(ab)=aj(b), a~A, be B. Since B is characteristic, B = h(B) =J(B) = B'. Thus it suffices to show, as we now do, that A C A'. Let ~zj:G ~ Hj (2 Im(i~)C (~ Ker(~,)=A' r=2 and hence that ACA'. Q.E.D. ProoJ oJ (2.1). Let ~=(Ai, Gi, Pi)l Gk ProoJ oJ P(1). Suppose that m>l. Put A=A 1, B={1}, A'=A',,, and B'-A'- m-1 o .... A1,' and apply Lemma(2.5). Incidentally, the whole point of introducing the class g+ is to allow the trivial factor B at this point. From (2.5) we get that B'=B= {1}, hence that A',= {1} for 1- Prooj that P(k-1) ~ P(k). Put A=A k, B=Ak_1O ... oAp A'=A'~, and B'=Am-1r ~ "" ~ t By Lemma (2.5), we get that B=B', so that, by induction, (k- 1)=(m- 1), and for all i, 1 Proo[ oJ (2.2). If G=Aa ..... A k where each A i is characteristic and centreless, then it follows by induction on (2.5) that the natural map k ~:l-I Aut(Ai)--+ Aut(G) i=l is an isomorphism. Thus if (Ai, G i, p~), _ The isomorphism k 1-[ Aut(Gi) S tT p, ~- Aut(G) i=1 follows from (1.1). Q.E.D. [Corollary (2.2)] Using results of Borel [4] and Margulis [29], one can construct large classes of discrete subgroups of Lie groups which satisfy (~.1), (~.2), and (~.3). Without attempting to be exhaustive, let !~ be the following class. !~ = {H :H is a discrete uniform subgroup in a noncompact connected simple adjoint Lie group L with ranks(L)>2.} Proposition 2.6. Any Jinite subclass of !~ satisfies (~3.1), (~.2), and (~.3). Pro@ Let 9t be the following relation on ~: H~RK iff H is isomorphic to a subgroup of K. Trivially ~t is reflexive and transitive. We also note that 9t is antisymmetric in the following strong sense: H~RK and K~RH ~ H~-K and any injective homomorphism i:H~K is an isomorphism. To see this, take an imbedding i(H) ~,H. If [H :i(H)] is infinite, let X be the symmetric space of L DH. By passing to a subgroup of finite index, H~Y and i(H)~g are manifolds of type K(H, 1) and dimension n=dim(X). But H n(H~, ~2) ~- :g2 since H~X is closed, and U,(i(H)~, 7Z2)~- H~ ~2) ~ 0 since i(H)~g is noncompact connected. Thus [H:i(H)] is finite, and the result follows from [4, Proposition 1.7]. Now let 35 = {H 1.... , H,} be a finite subclass of !e. Without loss of generality we may suppose that HigH s for i=i=j. Define h:35-+N by h(H)=O iff KglH ~ H~K. h(H)=n+l iff KiRH and HgK ~ h(K) 22 2=1 ) S~+ ~ = (c o, cL, ..., c, .CoC ~ ... c, S2+=(x1,...,x,,Yl,...,yn: FI xiYixZly71=I) i=1 Then SURFACEwFREE= {F., S,+- 1, Sz,.+:n>2}. The ordering we take is F2 - + (i) q~.$2,--'$2, (n>2), (ii) cp:Sz+--~F2, (n>2) or (iii) q~:S~--F, (n>3). To see (i), observe that an epimorphism Sz,---,S2,- + would induce an epimorphism H I(Sz,,-- . 7Z)--"H I(Sz,,Z).4- . However this is impossible since Hl(S2n;TZ)~TZzn-l~)(TZ/27Z) and Ha(Sz,,•)=7Z+.~ 2,. To see (ii), let ~p:F2,---S2+ be the epimorphism defined by the standard pre- sentation. Then the existence of an epimorphism ~o.S2,--,F~,. + would imply the existence of an epimorphism q~p:Fz---,F2, with nontrivial kernel, contradicting the Hopfian property of F2, [22, p. 109]. (iii) is similar to (ii). Thus to prove (~.3), it suffices to show that (iv) There is no nontrivial homomorphism O:H~K with O(H) fi Aut(Gi)~O'p, ~ , Aut(G) 11[ h [ 1 (I Out(c,)Io , Out( ). i=1 3. Weak Rigidity and Realisations of Extensions Let X be a smooth closed manifold, G(X) its monoid of homotopy equivalences with the compact open topology, and DIFF(X) its group of diffeomorphisms, topologised as in [20]. The inclusion ~p :DIFF(X)~G(X) induces a map between classifying spaces, B~p:BDIFF(X)--*BG(X), where BM is Milgram's classifying space for a topological monoid M [28]. DeJinition. A smooth closed manifold X is said to be weakly rigid iff B~p :B DIFF(X)~BG(X) admits a homotopy right inverse. 356 F.E.A.Johnson This definition is motivated by the following considerations. Let ~ = (E 1--,B) be a Hurewicz fibration over a CWcomplex B with fibre homotopy equivalent to X. is classified up to fibre homotopy equivalence by a homotopy class c(~):B~BG(X). Now suppose that X is weakly rigid, with s :BG(X)~B DIFF(X) a homotopy right inverse to Bq~. If ~ = s oc(~), then the following diagram commutes up to homotopy. BDIFF(X) ./1~ [ t~ B~o / c(~) = BG(X) Let q' be the locally trivial X bundle classified by & If, in addition, B is a smooth manifold, then by standard approximation arguments, we can replace t/' by an equivalent smooth fibre bundle q=(E2--*B) with fibre diffeomorphic to X. Note that q is still fibre homotopy equivalent to ~. Summarising, we obtain Proposition 3.1. Let X be a smooth closed weakly rigid manifold, B a smooth manifold, and ~ = (Et-~ B) a Hurewicz fibration whose fibre is homotopy equivalent to X. Then ~ is fibre homotopy equivalent to a smooth locally trivial fibre bundle t/=(E2-*B ) with fibre diffeomorphic to X. Definition. We say that a Poincar6 Duality (= PD) group G is smoothly realisable when there is a smooth closed aspherical manifold X G with nl(XG) = G. X G is called a smooth model for G. More generally, we say that an exact sequence of PD groups ~ =(K,-~F--~Q) is smoothly realisable when there exists a smooth locally trivial fibre bundle p :X~XQ with fibre X K, where X K, X r, XQ are smooth models for K, F, and Q respectively, and such that the long homotopy exact sequence of t/=(Xr, XQ, p, XK) reduces to ~. A PD group F is said to be weakly rigid with X r as model iff F has a smooth realisation with a weakly rigid manifold X r as model. Theorem 3.2. Let if: = (K~--.F---~Q) be a short exact sequence of PD groups in which (i) K and Q are smoothlyrealisable with models X~ and XQ respectively, and (ii) Xx is weakly rigid. Then has a smooth realisation with Xx, X Q as models for the fibre and base respectively, and in particular, F has a smooth realisation. Proof If H is a discrete group, let (H-*WH~WH) be the Eilenberg-Mac- lane principal simplicial H-bundle with contractible total space. Applying 17V to a short exact sequence ~=(K,---~F-oQ) yields a minimal Kan fibration W~=(WK--,WF~WQ) which in turn, upon taking geometrical realisations, yields a fibre bundle which is locally trivial in the category of compactly generated spaces [9, p. 55] thus ; IW-"~[ = ([ WK[ ~[ WFI---, [WQ[). Now suppose that there is a smooth manifold X a of homotopy type K(Q, 1) - [WQ[. Replacing the base IWQ[ of IW~l by XQ, we obtain a fibre bundle r ), which, since XQ is locally compact, is locally trivial in the category of (all) topological spaces, hence is Automorphisms of Direct Products of Groups 357 a Hurewicz fibration. Observe that the long homotopy exact sequence of ~ reduces to the original exact sequence ~. Finally suppose that ~=(K,--,F--~Q) is a short exact sequence of PD groups such that K and Q are smoothly realisable by X K and XQ respectively, and such that X K is weakly rigid. By (3.1), we may replace the fibration ~ constructed above by some smooth locally trivial fibre bundle t/=(Xr, XQ, p, Xr). In particular, F is smoothly realisable with X r as model. Q.E.D. At this point it is appropriate to observe that, for any K(F, 1) space X r, the homotopy type of BG(Xr) is quite simple to describe. Namely, it is the total space of a fibration (K(3, 2)~BG(Xr)-*BOut(F)), where 3 is the centre of F, Out(F) = Aut(F)/Inn(F) is the outer automorphism group ofF, and where the action of Out(F) on 3 is the obvious one, induced by evaluation of Aut(F) on F. In the case where F is centreless, BG(Xr) is homotopy equivalent to BOut(F). Compare [10]. Many geometrically interesting groups are weakly rigid. We give examples in Sects. 4 and 5. We end this section by proving some basic results on weak rigidity and smooth realisations. Proposition 3.3. Suppose that H, K are centreless weakly rigid groups with models X n and Xr, and suppose that the natural map ~ :Aut(H) x Aut(K)-*Aut(H x K) is an isomorphism. Then H x K is weakly rigid with X n x X K as model. Proof. Clearly ~ :Aut(H) x Aut(K)--~ Aut(H x K) induces an isomorphism :Out(H) x Out(K) -~ ,Out(H x K). Let j denote the canonical inclusion j :DIFF(Xn) x DIFF(XK)~DIFF(X n xXK). We get a homotopy commutative diagram as below BDIFF(Xu) x BDIFF(XK) i ,B(DIFF(XH ) x DIFF(XK) ) j ,BDIFF(X u xXK) BOut(H) x BOut(K) h , B(Out(H) x Out(K))' B(~ -'~ BOut(H x K), where the 2, are induced by the natural maps DIFF(Y)--*Out(nl(Y)) and where i and h are homotopy equivalences. By hypothesis, ,t t has a homotopy right inverse. Hence so also does 2 2. Let p be a homotopy right inverse for 2 2. Then v=jol~oB(~-1) is the required homotopy right inverse. Q.E.D. Proposition 3.4. Let F be a PD group with trivial centre, which is not decomposable as a nontrivial direct product, and which is smoothly realisable with X r as model. Suppose also that ~p :DIFF(Xr)~Out(F ) ~-G(Xr) is a homo topy equivalence. Then 358 F.E.A. Johnson for each n ~ 1, F ~") is weakly rigid with Xr,.,=Xr• ... xX r n as model. Proof First note that the hypothesis that (p:DIFF(Xr)~Out(F) is a homotopy equivalence implies that B~p:B DIFF(Xr)~BOut(F) is a homotopy equivalence, and hence that X r is weakly rigid. DIFF(Xr~.~ ) has an obvious subgroup S(F, n) defined as follows;f~S(F, n) iff there are diffeomorphisms (f)l = f(xl ..... Xn) = (fl (X~(1)..... f~(X~(n)) Clearly S(F, n)~-DIFF(Xr)~O" .. Indeed there is a commutative diagram DIFF(Xr)~O" . ~- , s(r,n) , i ,DIFF(Xr,.,) Out(r)Io- . ~ , Out(r~"~) in which ~ is an isomorphism by (1.1) and (2.9), and in which the vertical arrows are homotopy equivalences by virtue of the fact that the natural map q~ : DIFF (X r)~Out(F) is a homotopy equivalence. Let # : B 0 ut (F ("))--,BS(F, n) be a homotopy inverse for Bq~,:BS(F,n)~BOut(F(")).Then j=(Bi)# is the desired homotopy right inverse. Q.E.D. Recall that if 9.1 is a class of groups, then by a poly-9.1 filtration of length n on a group G we mean a sequence (Gr)o Proposition 3.5. A strongly poly-{weakly rigid} group is smoothly realisable. To conclude this section, we note that the concept of weak rigidity given here is different from, and supercedes, the now obsolete concept with the same name given in [14]. Automorphisms of Direct Products of Groups 359 4. Weak Rigidity of Products of Surface Groups Let F be a finite product of Surface groups. We may write F in the form F = F~"I)x... x F~"m~ where (Fi)l _~i_~,, is a sequence of Surface groups, and (ni)l~=i<= m is a sequence of positive integers, and F~ ~r Fj if i+j. Let Xr, be the smooth closed 2-manifold with ~I(Xr,)=F~. Then F is smoothly realisable with the following standard model for K(F, 1); x ... Proposition 4.1. Let F be a finite product of Surface groups. Then F is weakly rigid in its standard model. Proof. Adhering to the conventions above, write r=rp,>x ...... x F~nm, and Xr=Xtrn~'x xX rm Corollary 4.2. Let 1--*K--*F--*Q--* 1 be an exact sequence of PD groups in which K is a finite product of Surface groups, and Q is smoothly realisable. Then F is smoothly realisable. Proof Immediate from (3.2) and (4.1). Q.E.D. In [16] we showed that every poly-Surface group has a subgroup of finite index which is smoothly realisable. The problem of realising all poly-Surface groups is still outstanding however, and seems at present to be rather intricate. The problem begins in dimension six, that is, with poly-Surface length three : a poly-Surface group of length n has cohomological dimension 2n, and the four- dimensional groups were realised in [16]. To end this section, we use the results of Sects. 1-3 to realise some 6-dimensional groups neglected in [16]. Let __< be the ordering on SURFACE introduced in (2.7). Then the following is true. 360 F.E.A. Johnson Proposition 4.3. Let ~: (1 --* G~ i ~G . ~Q ~ 1) be an extension with G 1, Q Surface groups satisfying G 1 Hence O(G1) = Q. Put H 1 = i(G1) and H 2 = ~i(GI). Then Hi, H2< G and H 1 "H 2 = G. Also we have an exact sequence I__.HI~H2__~H2 v Q~I. If H1 c~H 2 4= {1}, then Q is a proper quotient of H 2 ---Q, contradicting the Hopfian property of the Surface group Q [8]. Hence G=HloH2~-G~xQ, and the extension ~ is trivial. An isomorphism ~:G 1 x GI~G is given by 7(x,y)=i(x).Ti(y). Q.E.D. Corollary 4.4. Let G be a poly-Surface group constructed from exact sequences as follows ; (i) l~Gl~G2~Ql~1 (ii) l~G2~G-~Q2~1, where G1, Q1, and Q2 are Surface groups with G 1 Proof Suppose that G~ is characteristic in G 2. Then G is strongly poly-Surface, hence is smoothly realisable by (3.5). If G~ is not characteristic, then by (4.3), G 2 ~ G~ x Gp so that G 2 is weakly rigid by (3.3), and G is smoothly realisable by (3.2). Q.E.D. This is an improvement on the results of [16], where we needed to assume that rank(G1) 5. Weak Rigidity of Discrete Uniform Subgroups In Sect. 4 we showed that finite products of Surface groups are weakly rigid. By Uniformisation Theory for complex curves, orientable Surface groups can be regarded as discrete uniform subgroups of the connected group PSL2(IR ). Non- orientable Surface'groups are discrete uniform subgroups of the two component group PGL2(~). Consequently, a finite product of Surface groups is a discrete uniform subgroup of a finite product of copies of PGL2(~). To conclude this paper, we investigate the weak rigidity of subgroups of more general Lie groups. To begin with, we note the following, which is, ignoring holomorphic arguments, the essential content of [5]. Automorphisms of Direct Products of Groups 361 Proposition 5.1 (Conner and Raymond [5]). 77" is weakly rigid with Z"\~,," as model. From now on, for simplicity, we concentrate on discrete uniform subgroups in connected semisimple Lie groups. Since we shall make heavy use of Mostow's Strong Rigidity theorem [24], to clarify matters we make the following definition. Definition. By a Mostow pair (G, F) we mean (i) a connected semisimple adjoint Lie group G, having no compact factors (ii) a torsion free discrete uniform subgroup F of G such that (iii) for any epimorphism n:G~PSLa(IR), n(F) is non-discrete in PSL2(IR). If (G, F) is a Mostow pair, we put X r = F\G/K where K is a maximal compact subgroup of G. Our first step is to show that F is weakly rigid with X r as model. This is such a routine consequence of Mostow's theorem that we may reasonably attribute it to Mostow. However, since the proof does not seem to be written down anywhere, we give it below. To begin with, we need the following which, though presumably very well known, likewise seems to lack a convenient reference in the literature. Lemma 5.2. Let G be a connected semisimple noncompact Lie group, and let K be a maximal compact subgroup. Then K is self-normalising. Proof Let g = fop be the corresponding Cartan decomposition of the Lie algebra g of G. The Killing form B 0 of g is positive definite on p and negative definite on f. First suppose that y~p centralises f. We show that y=0. For [y,p]Cf since [p, p] C f, and I-y,f] = 0. Thus ad(y) ad(y) = 0 so that Bo(y, y) = Tr(ad(y) 2) = 0. Hence y=0. Now let x~ N0(f ) = {z~ 9 : [z, f] C f}. Write x = x o + Xl, Xo~: f, X 1E p. Since If, {] C f and If, p] Cp then x 1 centralises f so that x 1 =0 by above. Hence X=Xo~f. Thus f = N0(f ). Consequently K = No(K ) = {g6 G :gKg- 1 __ K}. Q.E.D. Theorem 5.3. Let (G, F) be a Mostow pair. Then F is weakly rigid with X r = F\G/K as model, where K is a maximal compact subgroup of G. Proof Put X = G/K, Xr=F~, and let x= [1] =KeG~K, so that K=Gx={g~G:g.x=x }. Ifee Aut(G) then ~(K) is a maximal compact subgroup of G, and since any two maximal compact subgroups are conjugate, we may choose h~ G such that c~(K) = h~Kh] a. We define a diffeomorphism ~, :X~X by means of 0r x) = ~(g)" h~" x. This definition does not depend on the particular representation of a point in X in the form g.x. No more does it depend on the particular choice of h, ~ G. For if h~Kh~ 1 = k, Kk~ 1 then since K is self-normalising, k, = h,. k for some ke K. Thus or since k.x=x. Moreover, it is straightforward to see that (aft), =%fl,. Hence we have a homomorphism Aut(G)~Diff(X), ~0r Now appeal to Mostow's theorem. Under our hypotheses, an automorphism ~:F~F extends uniquely to an automorphism ~ : G~G. Thus we get a homomor- phism Aut(F)-~Diff(X) by means of ~(~),. This in turn gives rise to a 362 F.E.A. Johnson homomorphism ~v : Aut(F)~Diff(Xr) as follows : ~v(ct)[F-g. x] = F.( E),(g. x) = F. E(g). h~. x. Finally note that if ~eInn(G) then ~v(~)=Id. For if cr zeF, we may take h~ = z so that ~(~)[F.g.x]=F.z.g.x=F.g.x, since zeF. It is now easy to check that the homomorphism ~p,:Out(F)~DIFF(Xr) induced by ~p is a right inverse to the canonical map q~:DIFF(Xr)~no(G(Xr) ) =Out(F). Since F is centreless (see Corollary (5.18), p. 84 of [26]), BOut(F) is homotopy equivalent to BG(Xr)), so that the induced map B~. :BG(Xr) ) ~- B Out (F)~B DIFF(Xr) is the required homotopy right inverse. Q.E.D. Of the two approaches we have given to weak rigidity, one uses low dimensional techniques, generalisation of which to higher dimensions seem very difficult, and one uses Mostow's Strong Rigidity theorem, which works well except in low dimensions, precisely except when G has a PSL2(IR) factor in which the projection of F is discrete, in which case it is known to be false. Let us examine how far we are from being able to treat the general case. Recall [26, p. 86] that if G is a connected adjoint semisimple Lie group and P is a discrete uniform subgroup of G then P is called irreducible iff n(P) is non-discrete for any projection n of G onto a proper direct factor. In general, if P is a discrete uniform subgroup in such a G, then G admits a direct product decomposition G= G 1 ..... G m into connected normal subgroups G i, such that Pi=Gic~P is irreducible in G~, and such that P0 = P1 o... op,, has finite index in P. G~ need not be simple. By collecting factors, we may re-express this as follows, separating factors G i with ~,-rank_>2 from factors with ~,-rank = 1 ; P has a normal subgroup Po of finite index in P such that Po is a direct product Po = H x H' x H" where H is a product of irreducible discrete uniform subgroups in factors G i with IR-rank(Gi)__> 2, H' is a product of Surface groups, and H" is a finite product L 1 x ... x Lp, where each L i is a discrete uniform subgroup in a connected adjoint Lie group of lR-rank 1 which is not isomorphic to PSL2(F,). That is, L~ is a discrete uniform subgroup in some PSO(n, 1) (n>3), PSU(n, 1) (n>2), PSp(n, 1) (n=> 1), or in the isometry group of the noncompact symmetric space (f4(-20), ~O(9)). Then P/Po and H" measure the deviation of our hypotheses from the ideal in the following sense. Theorem 5.4. Let P= H x H' be a product group in which the factors satisfy the following : (i) (G, H) is a Mostow pair for some G such that n(H) is non-discrete for any epimorphism n : G~ L onto a Lie group L with lR-rank(L) = 1, and (ii) H' is a finite product of Surface groups. Then P is weakly rigid. In particular, for any exact sequence ~=(I~HxH'~G~Q~I), if Q is smoothly realisable, then so is G. Automorphisms of Direct Products of Groups 363 Proof. The hypothesis on H in (i) above is equivalent to saying that in the decomposition of a subgroup H 0 of finite index in H, into irreducible factors, each of the irreducible factors is irreducible in a Lie group of R-rank > 2. Let q:P--,H' be the projection. Observe that any non-trivial subgroup A of H' has an infinite cyclic summand in H1(A :2~). This is obvious by projection of A into the various Surface group factors of H'. 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