We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the G-CW-version EF(G) and the numerable G-space version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C ∗-algebra, for the Farrell-Jones Conjecture about the algebraic K-and L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.

Abstract. Let G be a word hyperbolic group in the sense of Gromov and P its associated Rips complex. We prove that the fixed point set P H is contractible for every finite subgroup H of G. This is the main ingredient for proving that P is a finite model for the universal space EG for proper actions. As a corollary we get that a hyperbolic group has only finitely many conjugacy classes of finite subgroups.

Abstract. We redefine the Baum-Connes assembly map using simplicial approximation

We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings of D-branes which generalize previous examples. We propose a definition for groups of differential characters associated to equivariant K-theory. We derive a Dirac quantization rule for Ramond-Ramond fluxes, and study flat Ramond-Ramond potentials on orbifolds.

Abstract. We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups and compute these via a classifying space and as K-theory groups of suitable σ-C ∗-algebras. We also relate equivariant vector bundles to these σ-C ∗-algebras and provide sufficient conditions for equivariant vector bundles to generate representable K-theory. We mostly work in the generality of locally compact groupoids with Haar system. 1.

Let G be countable group and M be a proper cocompact even-dimensional G-manifold with orbifold quotient ¯ M. Let D be a G-invariant Dirac operator on M. It induces an equivariant K-homology class [D] ∈ KG 0 (M) and an orbifold Dirac operator ¯ D on ¯ M. Composing the assembly map KG 0 (M) → K0(C ∗ (G)) with the homomorphism K0(C ∗ (G)) → Z given by the representation C ∗ (G) → C induced from the trivial representation of G we define index([D]) ∈ Z. In the second section of the paper we show that index ( ¯D) = index([D]) and obtain explicit formulas for this integer. In the third section of the present paper we review the decomposition of KG 0 (M) in terms of the contributions of fixed point sets of finite cyclic subgroups of G obtained by W. Lück. In particular, the class [D] decomposes in this way. In the last section we derive an explicit formula for the contribution to [D] associated

Abstract. We introduce a new construction, the isotropy groupoid, to organize the orbit data for split Γ-spaces. We show that equivariant principal G-bundles over split Γ-CW complexes X can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex A = Γ\X is a graph, with all edge stabilizers toral subgroups of Γ, we obtain a purely combinatorial classification of bundles with structural group G a compact connected Lie group. If G is abelian, our approach gives combinatorial and geometric descriptions of some results of Lashof-May-Segal [18] and Goresky-Kottwitz-MacPherson [10].

Orbispaces are spaces with extra structure. The main examples come from topological group actions X G and are denoted [X/G], their underlying space, or coarse moduli space being X/G. By definition, every orbispace is locally of the form [X/G], but the group G might vary. We shall work with orbispaces whose coarse moduli spaces are CW-complexes, and whose stabilizer groups are compact Lie groups. We also require the stratification of the coarse moduli space by the type of stabilizer group to be compactible with the CW structure. A convenient model for an orbispace is then given by a topological groupoid (see [2], [3]). An orbispace always comes with a map to its coarse moduli space. By a suborbispace X ′ ⊂ X, we shall mean an orbispace obtained by pulling back along a subspace of the coarse moduli space. If X is an orbispace modeled by a topological groupoid G, then a vector bundle over X is a vector bundle over the space of objects of G equipped with an action of the arrows of G. It is tempting to define K-theory as the Grothendieck

Abstract. Let G be a word hyperbolic group in the sense of Gromov and P its associated Rips complex. We prove that the fixed point set P H iscontractible for every finite subgroup H of G. Thisisthe main ingredient for proving that P is a finite model for the universal space EG for proper actions. As a corollary we get that a hyperbolic group hasonly finitely many conjugacy classes of finite subgroups.

Abstract. In this paper we define twisted equivariant K-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid G, this defines a periodic cohomology theory on the category of finite G-CW-complexes with G-stable projective bundles. A classification of these bundles is shown. We also obtain a completion theorem and apply these results to proper actions of groups. Key words: Twisted K-theory, groupoids, proper actions, completion theorem.