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.

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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...