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Algebraic K-Theory of Groups Wreath Product with Finite Groups

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Algebraic K-Theory of Groups Wreath Product with Finite Groups View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 154 (2007) 1921–1930 www.elsevier.com/locate/topol Algebraic K-theory of groups wreath product with finite groups S.K. Roushon 1 School of Mathematics, Tata Institute, Homi Bhabha Road, Mumbai 400005, India Received 22 June 2006; received in revised form 31 January 2007; accepted 31 January 2007 Abstract The Farrell–Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for sev- eral classes of groups. For example, for discrete subgroups of Lie groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249–297], virtually poly-infinite cyclic groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249–297], Artin braid groups [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515–526], a class of virtually poly-surface groups [S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press] and virtually solvable linear group [F.T. Farrell, P.A. Linnell, K-Theory of solv- able groups, Proc. London Math. Soc. (3) 87 (2003) 309–336]. We extend these results in the sense that if G is a group from the above classes then we prove the conjecture for the wreath product G H for H a finite group. The need for this kind of extension is already evident in [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515–526; S.K. Roushon, The Farrell–Jones isomorphism conjecture for 3-manifold groups, math.KT/0405211, K-Theory, in press; S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press]. We also prove the conjecture for some other classes of groups. © 2007 Elsevier B.V. All rights reserved. MSC: primary 19D55; secondary 57N37 Keywords: Fibered isomorphism conjecture; 3-Manifold groups; Discrete subgroup of Lie group; Poly-Z groups; Braid groups 1. Introduction In this article we are mainly concerned about the Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory. We extend some existing results and also prove the conjecture for some other classes of groups. Finally we deduce a corollary for the Isomorphism Conjecture for the algebraic K-theory in dimension 1(see Corollary 2.3). E-mail address: [email protected]. URL: http://www.math.tifr.res.in/~roushon/paper.html. 1 This work was done during the author’s visit to the Mathematisches Institut, Universität Münster under a Fellowship of the Alexander von Humboldt-Stiftung. 0166-8641/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2007.01.019 1922 S.K. Roushon / Topology and its Applications 154 (2007) 1921–1930 Before we state the theorem let us recall that given two groups G and H the wreath product G H is by definition the semidirect product GH H where the action of H on GH is the regular action. Theorem 1.1. The Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory is true for the group G H where H is a finite group and G is one of the following groups. (a) Virtually polycyclic groups. (b) Virtually solvable subgroups of GLn(C). (c) Cocompact discrete subgroups of virtually connected Lie groups. (d) Artin full braid groups. (e) Weak strongly poly-surface groups (see Definition 4.1). (f) Extensions of closed surface groups by surface groups. (g) π1(M) Z, where M is a closed Seifert fibered space. Proof. See Corollary 2.2 and Remark 2.1. 2 ? P H∗ (In the notation defined in Definition 2.1 the theorem says that the FICVC is true for G H or equivalently the ? P H∗ FICwFVC is true for G.) ? P H∗ The FICwFVC was proved for 3-manifold groups in [12] and [13] and for the fundamental groups of a certain class of graphs of virtually poly-cyclic groups in [14]. ? P H∗ For the extended Fibered Isomorphism Conjecture FICwFVC we deduce the following proposition using (a) of ? ? P H∗ P H∗ Lemma 3.4. This result is not yet known if we replace FICwFVC by FICVC (see 5.4.5 in [9] for the question). ? P H∗ Proposition 1.1. Let G be a group containing a finite index subgroup Γ . Assume that the FICwFVC is true for Γ . ? P H∗ Then the FICwFVC is true for G. We work in the general setting of the conjecture in equivariant homology theory (see [1]). We find out a set of properties which are all satisfied in the pseudoisotopy case of the conjecture. And assuming these properties we prove a theorem (Theorem 2.2) for the Isomorphism Conjecture in equivariant homology theory. Theorem 1.1 is then a particular case of Theorem 2.2. We mention in Corollary 2.3 another consequence of our result for the Isomorphism Conjecture in the algebraic K-theory case. We also hope that the general Theorem 2.2 will be useful for future application. The methods we use in this paper were developed in [14]. 2. Statement of the general theorem We first recall the general statement of the Isomorphism Conjecture in equivariant homology theory from [1] and also recall some definitions from [14]. AclassC of subgroups of a group G is called a family of subgroups if C is closed under taking subgroups and conjugations. For a family of subgroups C of G, EC(G) denotes a G-CW complex so that the fixpoint set EC(G)H is contractible if H ∈ C and empty otherwise. And R denotes an associative ring with unit. ? (Fibered) Isomorphism Conjecture 1. (See Definition 1.1 in [1].) Let H∗ be an equivariant homology theory with values in R-modules. Let G be a group and C be a family of subgroups of G. Then the Isomorphism Conjecture for the pair (G, C) states that the projection p : EC(G) → pt to the point pt induces an isomorphism HG HG HG n (p) : n EC(G) n (pt) ∈ Z ∈ Z HG for n .AndtheIsomorphism Conjecture in dimension k,fork , states that n (p) is an isomorphism for n k. Finally the Fibered Isomorphism Conjecture for the pair (G, C) states that given a group homomorphism φ : K → G the Isomorphism Conjecture is true for the pair (K, φ∗C).Hereφ∗C ={H<K| φ(H)∈ C}. S.K. Roushon / Topology and its Applications 154 (2007) 1921–1930 1923 From now on let C be a class of groups which is closed under isomorphisms, taking subgroups and taking quotients. The class VC of all virtually cyclic groups has these properties. Given a group G we denote by C(G) the class of subgroups of G belonging to C. Then C(G) is a family of subgroups of G which is also closed under taking quotients. Definition 2.1. (See Definition 2.1 in [14].) If the (Fibered) Isomorphism Conjecture is true for the pair (G, C(G)) ? ? ? H∗ H∗ H∗ we say that the (F)ICC is true for G or simply say (F)ICC (G) is satisfied. Also we say that the (F)ICwFC (G) is ? H∗ satisfied if the (F)ICC is true for G H for all finite groups H . ? ? Let us denote by P H∗ and K H∗ the equivariant homology theories arise in the Isomorphism Conjecture of Farrell and Jones [4] corresponding to the stable topological pseudoisotopy theory and the algebraic K-theory, respectively. For these homology theories the conjecture is identical with the conjecture made in Sections 1.6 and 1.7 in [4]. (See Sections 5 and 6 in [1] for the second case and 4.2.1 and 4.2.2 in [9] for the first case.) ? ? ? ? H∗ H∗ H∗ H∗ Note that if the FICC (respectively, FICwFC ) is true for a group G then the FICC (respectively, FICwFC )is ? H∗ true for subgroups of G. We refer to this fact as the hereditary property. Also note that the (F)ICC is true for H ∈ C. ? H∗ Definition 2.2. (See Definition 2.2 in [14].) We say that the PC -property is satisfied if for G1,G2 ∈ C the product ? H∗ G1 × G2 satisfies the FICC . In the following the notation A B stands for the semidirect product of A by B with respect to some arbitrary action of B on A. Definition 2.3. We define the following notations. ? H∗ N The FICwFVC is true for π1(M) for closed nonpositively curved Riemannian manifolds M. ? H∗ n B The FICwFVC is true for Z Zfor all n. ? ? H∗ H∗ L If G = limi→∞ Gi and the FICVC is true for Gi for each i then the FICVC is true for G. ? P H∗ Theorem 2.1. B, N and L are satisfied for the FICwFVC . Proof. See Theorem 4.8 in [4] for B, Proposition 2.3 in [4] and Fact 3.1 in [6] for N and Theorem 7.1 in [5] for L. 2 ? H∗ Theorem 2.2. (1) B implies that the FICwFVC is true for the following groups. (a) Finitely generated virtually polycyclic groups. (b) Finitely generated virtually solvable subgroup of GLn(C). In addition if we assume L then we can remove the condition ‘finitely generated’ from (a) and (b).
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