Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G/H, considered as left G-sets, and morphisms are G-maps. This is a useful construct for organizing the study of fixed sets and quotients of G-actions. If G acts on a set X, there is the contravariant fixed point functor Or(G) −→ SETS given by G/H → X H = map G (G/H, X) and the covariant quotient space functor Or(G) −→ SETS given by G/H → X/H = X × G G/H. Bredon used the orbit category to define equivariant cohomology theory and to develop equivariant obstruction theory.
Abstract. We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic FarrellJones conjecture can be reduced to only those virtually cyclic groups which admit a surjection with finite kernel onto a cyclic group. The second is that the Waldhausen Nil groups for a group which maps epimorphically onto the infinite dihedral group can be computed in terms of the Farrell-Bass Nil groups of the index two subgroup which maps surjectively to the infinite cyclic group.
Abstract. Let Γ be a semidirect product of the form Z n ⋊ρ Z/p where p is prime and the Z/p-action ρ on Z n is free away from the origin. We will compute the topological K-theory of the real and complex group C * -algebra of Γ and show that Γ satisfies the unstable Gromov-Lawson-Rosenberg Conjecture. On the way we will analyze the (co-)homology and the topological K-theory of the classifying spaces BΓ and BΓ. The latter is the quotient of the induced Z/p-action on the torus T n . IntroductionLet p be a prime. Let ρ : Z/p → Aut(Z n ) = GL(n, Z) be a group homomorphism. Throughout this paper we will assume:Condition 0.1 (Free conjugation action). The induced action of Z/p on Z n is free when restricted to Z n − 0.Denote bythe associated semidirect product. Since Γ has a finitely generated, free abelian subgroup which is normal, maximal abelian, and has finite index, Γ is isomorphic to a crystallographic group. An example of such group Γ is given by Z p−1 ⋊ ρ Z/p where the action ρ is given by the regular representation Z[Z/p] modulo the ideal generated by the norm element. When n = 1 and p = 2, Γ is the infinite dihedral group.Let BΓ := Γ\EΓ be the classifying space of Γ. Denote by EΓ be the classifying space for proper group actions of Γ. Let BΓ = Γ\EΓ. The space BΓ is the quotient of the torus T n under the Z/p-action associated to ρ. It is not a manifold, but an orbifold quotient.To compute the K-theory of the C * -algebra, we will use the Baum-Connes Conjecture which predicts for a group G that the complex and real assembly mapsare bijective for n ∈ Z. The point of the Baum-Connes Conjecture is that it identifies the very hard to compute topological K-theory of the group C * -algebra of G to the better accessible evaluation at EG of the equivariant homology theory given by equivariant topological K-theory. The Baum-Connes Conjecture has been proved for a large class of groups which includes crystallographic groups (and many . We will later use the composite maps, where in each case the second map is induction with the projection Γ → {1}.Next we describe the main results of this paper. We will show in Lemma 1.9 (i) that k = n/(p − 1) is an integer. Let P be the set of conjugacy classes {(P )} of finite non-trivial subgroups of Γ.is always a finitely generated free abelian group. (ii) There is an exact sequenceRemark 0.4 (Twisted group algebras). The computation of Theorem 0.3 has already been carried out in the case p = 2 and in the case n = 2 and p = 3 in [17, Theorem 0.4, Example 3.7]. In view of [17, Theorem 0.3] the computation presented in this paper yields also computations for the topological K-theory K * (C * r (Γ, ω)) of twisted group algebras for appropriate cocycles ω. One may investigate whether the whole program of [17] can be carried over to the more general situation considered in this paper.Remark 0.5 (Computations by Cuntz and Li). Cuntz and Li [13] compute the K-theory of C * -algebras that are associated with rings of integers in number fields. They have to make the assumption that the alge...
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