No abstract

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.

No abstract

We prove the Borel Conjecture for a class of groups containing wordhyperbolic groups and groups acting properly, isometrically and cocompactly on a finite-dimensional CAT(0)-space.

Let X be a finite connected CW -complex. Suppose that its fundamental group π is residually finite, i.e., there is a nested sequence . . . ⊂ Γ m+1 ⊂ Γ m ⊂ . . . ⊂ π of in π normal subgroups of finite index whose intersection is trivial. Then we show that the p-th L 2 -Betti number of X is the limit of the sequence b p (X m )/[π : Γ m ] where b p (X m ) is the (ordinary) p-th Betti number of the finite covering of X associated with Γ m .

Let F ⊆ SL 2 (Z) be a finite subgroup (necessarily isomorphic to one of Z 2 , Z 3 , Z 4 , or Z 6 ), and let F act on the irrational rotational algebra A θ via the restriction of the canonical action of SL 2 (Z). Then the crossed product A θ ⋊α F and the fixed point algebra A F θ are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of Z 2 on any simple d-dimensional noncommutative torus A Θ . Along the way, we prove a number of general results which should have useful applications in other situations.1 2 ECHTERHOFF, LÜCK, PHILLIPS, AND WALTERS Theorem 0.1 (Theorems 4.9, 6.3, and 6.4). Let F be any of the finite subgroups Z 2 , Z 3 , Z 4 , Z 6 ⊆ SL 2 (Z) with generators given as above and let θ ∈ R Q. Then the crossed product A θ ⋊ α F is an AF algebra. For all θ ∈ R we haveFor F = Z k for k = 2, 3, 4, 6, the image of K 0 (A θ ⋊ α F ) under the canonical tracial state on A θ ⋊ α F (which is unique) is equal to 1 k (Z+θZ). As a consequence, A θ ⋊ α Z k is isomorphic to A θ ′ ⋊ α Z l if and only if k = l and θ ′ = ±θ mod Z.

We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C * -algebra and the Farrell-Jones Conjecture about the algebraic K-and L-theory of the group ring of a (discrete) group G.

We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic n-space H n with finite volume orbit space. We then apply this result to show that, for any Bianchi group Γ, W h(Γ), ˜ K 0 (ZΓ), and K i (ZΓ) vanish for i ≤ −1.