Algebraic Controlled K-Theory of Integral Group Ring of SL(3, Z)
S. Upadhyay1 School of Mathematics, Tata Institute of Fundamental Research, Bombay400005, India Abstract. We calculate the lower Controlled Algebraic K-theory of any nitely generated innite subgroup of SL(3, Z), the group of 3 3 integral matrices of determinant 1. Key words. Topological pseudo-isotopy, Quinns spectral sequence, Virtually (innite) cyclic groups.
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Introduction
Let SL(3, Z) denote the group of 3 3 integral matrices of determinant 1, and denote any nitely generated innite subgroup of SL(3, Z). In this paper we calculate the lower controlled algebraic K-theory of Z as dened by Farrell and Jones (cf. [4]). We refer the reader to [4] (Section 1 and Appendix) for various denitions. Let X denote any CW-complex and let C(X) denote the class of virtually
Current Address: Department of Mathematical Sciences, SUNY at Binghamton, Binghamton, NY-13902-6000, U. S. A.
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cyclic subgroups of 1 (X). A group G is virtually cyclic if it is either a nite group, or there is an exact sequence 0 T G F 0 where T is an innite cyclic group and F is a nite group. Let A denote a universal (1 (X), C(X))- space and X be the universal covering space of X. (See [4] , p. 250 for the denition of universal (1 (X), C(X))- space .) Let 1 (X)XA XA denote the diagonal action, then : E(X) B(X) and f : E(X) X are dened to be the quotient of the standard projections X A A and X A X under the 1 (X)- actions. Let F () denote any of the -spectra valued functors as in [4] (p. 251-253). Then Farrell and Jones conjecture that, F (f )oA : H (B(X), F ()) F (X) is an equivalence of the - spectra, where A is an assembly map for the simplicially stratied bration : E(X) B(X) and F (f ) is the image of f : E(X) X under the functor F (). Farrell and Jones verify this conjecture for the functor F () = P (), the topological pseudoisotopy spectrum (cf. [4], Th. 2.1) when 1 (X) is a co-compact discrete subgroup of a virtually connected Lie group. In this paper we calculate Hi (B(X), F ()), where i = 1, 0 and 1, for the case when X = K(, 1) and F () = P+2 (). Our main result is the following in which T k denotes the free abelian group of rank k. Theorem 1.1 For a nitely generated innite subgroup of SL(3, Z) and F () = P+2 (), Hi (E/, F (p)) vanishes for i = 0, 1, and is T k for i = 1, where k is the number of distinct conjugacy classes of D6 (the dihedral group of order 12) in . In particular, for = SL(3, Z), Hi (E/, F (p)) vanishes for i = 0, 1, and is innite cyclic for i = 1. We prove the above theorem for the special case when = SL(3, Z); the proof of the general case will be evident from this. Because of the simpler nature of the virtually (innite) cyclic subgroups of SL(3, Z), we rst reduce the problem of calculating H (B(X), F ()) to calculating H (B (X), F ( )), where B (X) and are dened as above except that instead of C(X), we now consider C (X), which is the class of nite subgroups of SL(3, Z). Also , A is 2
replaced by A which is a universal (1 (X), C (X))- space. Now using an explicit description of the space A as constructed by Soule (cf. [13]), complete determination of nite subgroups of SL(3, Z) up to conjugacy by Tahara (cf. [14]) and the knowledge of K-theory of nite groups which occur in SL(3, Z) (cf. [2], [10], [12]), we complete the calculation.
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Reduction to nite subgroups
We again refer the reader to [4] for various denitions. Let be a discrete group and C and C be two full classes of subgroups of with C C. Let E and E be universal (, C) and (, C )- spaces respectively and let E be a universal -space. Let AC : H (E/, F (p)) F (E/) AC : H (E /, F (p )) F (E/) AC ,C : H (E /, F (p )) H (E/, F (p)) be as dened in [4]. Here p,p denote the simplicially stratied brations E E E/, E E E / respectively. For S C, dene C S to be the class of subgroups of S such that G C S if and only if G C and G S and let E S be a universal (S, E S )- space. Then AC S : H (E S /S, F (p S ) F (ES /S) is similarly dened. We state the following theorem from [4] (cf. [4], Th. A. 10). Theorem 2.1 The relative assembly map AC ,C is an equivalence of -spectra, provided for each S C we have that AC S is an equivalence of -spectra. We formulate a lemma for our purpose. Lemma 2.2 The relative assembly map AC ,C is an isomorphism at the Hi level for all i 0 and surjective for i = 1, provided AC S is an isomorphism at the Hi -level for all i 1.
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Proof:
Follows from the proof of the Theorem 2.1 and the Five-lemma.2
Before we check the hypothesis of the above lemma for = SL(3, Z), we recall Quinns spectral sequence (cf. [4], Lemma 1.4.2, [11], Appendix). Lemma 2.3 Let f : E X be a simplicially stratied bration. Then there is a spectral sequence with E 2 p,q = Hp (X, q F (f )) which abuts to Hp+q (X, F (f )). Here j F (f ) denote the stratied system of groups {j F (f 1 (x)) : x X} over X. Now, let = SL(3, Z) and C and C denote class of virtually cyclic and nite subgroups of respectively. Let F () = P+2 () where P () is topological pseudoisotopy functor (cf. [6] for the denition of P ()). Note that by Anderson-Hsiangs result (cf. [1]) i P+2 (X) = Ki (Z(1 X)) for i 1, K0 (Z(1 X)) for i = 0, and W h(1 X) for i = 1. We will check the hypothesis of Lemma 2.2 in this case. If S C is nite subgroup of , then C S = all subgroups of S; hence E S can be taken to be a point. This yields that AC S is a weak homotopy equivalence; in particular AC S induces an isomorphism for all i when S C is nite. Now let S C be a virtually innite cyclic subgroup of . But any virtually innite cyclic subgroup is either of the type | F T where F is a nite subgroup of , or it maps onto D , the innite dihedral group (cf. [5], Lemma 2.5). (Here T denotes innite cyclic group | and F T denotes semi-direct product of F and T where T acts on F by an automorphism of F .) Using this fact and classication of nite subgroups of (cf. [14]), an elementary calculation shows that following are all the virtually innite cyclic subgroups of up to isomorphism
| T, T T2 , D , D T2 , T T4 .
()
Here Tn denotes nite cyclic group of order n and the action of on T is non-trivial. The above classication of virtually innite cyclic subgroup of | is obtained as follows. First, to identify the groups of the type F T | which occur in we observe that since F is a nite group, F T contains F nT for some positive integer n. Now, a case by case checking yields that for F a non-trivial nite subgroup of (which is completely classied up to conjugacy by Tahara cf. [14]) , the centralizer of F contains an element of innite order only if F = T2 . Next, to identify the groups which map onto 4
D , we proceed as follows. Let S map onto D with non-trivial kernel F , i.e.; we are given following exact sequence, 0 F S D 0. Now, this exact sequence gives rise to another exact sequence 0 F S T 0 where T is innite cyclic subgroup of index two in D and S is just the inverse image of T under the map S D . The sequence 0 F S T 0 splits since T is free . Hence F is isomorphic to T2 and S | is isomorphic T to T2 by the classication of the groups of the type F T in . Since S is a subgroup of index two in S, we have the following exact sequence 0 S S T2 0. An elementary argument shows that in the above exact sequence S has to be isomorphic to one of the following groups
| T T2 , T T2 T2 , D T2 , T T4
The rst two groups in the above list are ruled out since S has to map onto D . And it is a fact that the other two groups in the above list do occur as subgroups of . It is also a fact that Ki (ZF ) (for i 1),K0 (ZF ) and W h(F ) all vanish when F is any nite group occuring as a subgroup of any of the groups in the list (). This fact is proven in [2], [12], [10], [8]. A second fact is that Ki (ZS) (for i 1),K0 (ZS) and W h(S) also all vanish where S is any one of the groups in the list (). This is well known for the rst two groups in this list (cf. [7], p. 43,Remark ). The vanishing of Ki (ZS) (for i 1) for all groups in the list follows from [5] (cf. Th. 2.1) using the rst fact. The argument showing that K0 (ZS) = W h(S) = 0 for the last three groups in the list () is the following and uses [5] (cf. Th. 2.6) together with Lemma 2.3 applied to the simplicially stratied bration E : E X of [5] (cf. Sect. 2). The key point is that the fundamental group G of a ber of E is either the group T T2 or some nite subgroup of S. In either case Ki (ZG) for i 1,K0 (ZG) and W h(G) all vanish. Hence, the spectral 5
sequence shows that Hi (X, F (E )) = 0 for i = 0, 1. But [5] (cf. Th 2.6) says that these groups maps onto K0 (ZS) and W h(S) respectively. Once again using the spectral sequence of Lemma 2.3 together with the above two facts, we see that AC S is an isomorphism for all i 1. The hypothesis of Lemma 2.2 is consequently satised. We therefore conclude the following when = SL(3, Z),F () = P+2 () and the classes C and C are as above. Proposition 2.4 The relative assembly map AC ,C is an isomorphism at the Hi -level for all i 0 and surjective for i = 1.
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Calculation of Hi(E /, F(p ))
In this section we calculate Hi (E /, F (p )) for F () = P+2 () and 1 i 1. Here E is a universal (, C )- space, C is the class of nite subgroup of = SL(3, Z) and p is the simplicially stratied bration as described earlier. In [13] Soule constructs a simplicial complex X3 on which acts simplicially with all the istropy group being nite groups. We now check that the xed point set for any nite group in is non-empty and contractible. Using Soules notation, let X1 denote the space of positive denite symmetric matrices mod scalars. Since X1 can be given a Riemannain metric of nonpositive curvature, and acts on it by isometries, we see that the xed point set in X1 for any nite subgroup of is non-empty by Cartans theorem (cf. [9], p. 75, Theorem 13.5). It is in fact contractible (cf. [9], p. 82, Theorem 14.6). Since X3 is a retract of X1 by a -equivariant retraction (cf. [13]), we conclude that the xed point set in X3 for any nite subgroup of is non-empty and contractible. Hence, using a result of Connolly and Kozniewski (cf. [3]), we can identify E with the space X3 as constructed by Soule (cf. [13]). We rst calculate H0 (E /, F (p )) . From Lemma 2.3 there is spectral sequence with E 2 s,t = Hs (E /, t F (p )) with s+t = 0 which converges to H0 (E /, F (p )). By Carters results (cf. [2]), Hs (E /, t F (p )) = 0 if t 2. Hence the possible terms of the spectral sequence with E 2 s,t = 0 and s + t = 0 are E 2 0,0 = H0 (E /, 0 F (p )) and E 2 1,1 = H1 (E /, 1 F (p )). But these groups vanish as well since in the triangulation of the space E the groups which occur as the isotropy group of vertices under the action of are D2 , D4 , D6 , S3 and S4 . (Here Dn denote the dihedral group of order 2n 6
and Sn denotes the symmetric group on n letters.) For all of them, K0 vanishes(cf. [12]). Furthermore, the groups which occur as the stabilizer of a 1- simplex of E under the action of are D2 , D4 and S3 . For these groups, K1 vanishes as well (cf. [2]). It follows that H0 (E /, F (p )) = 0. While calculating H1 (E /, F (p )), we note as above that the only possibly non-zero term of the spectral sequence is E 2 0,1 = H0 (E /, 1 F (p )). Now since the groups which occur as stabilizer of 1-simplices of E under the action of are ...