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Abstract. We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A ⊂ M, the Betti numbers of the standard equivalence relation associated with A ⊂ M ([G2]), are in fact isomorphism invariants for the factors M, β HT n (M), n ≥ 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying β HT n (Mt) = β HT n (M)/t, ∀t> 0, and a Künneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z2 ⋊ SL(2, Z)), for which β HT 1 (M) = β1(SL(2, Z)) = 1/12. Thus, Mt ̸ ≃ M, ∀t ̸ = 1, showing that the fundamental group of M is trivial. This solves a long standing problem of R.V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.

1 Transverse measure for flows 4 2 Transverse measure for foliations 6

this paper we shall prove the coarse Baum-Connes conjecture for proper metric spaces with nite asymptotic dimension. Combining this result with a certain descent principle we obtain the following application to the Novikov conjecture on homotopy invariance of higher signatures. Theorem 1.1 Let be a nitely presented group whose classifying space B

Abstract. We give a useful classification of the metabelian unitary representations of π1(MK), where MK is the result of zero-surgery along a knot K ⊂ S 3. We show that certain eta invariants associated to metabelian representations π1(MK) → U(k) vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We compare this sliceness obstruction to the Casson–Gordon obstruction and show it is at least as strong. It turns out that eta invariants can in many cases be easily computed for satellite knots. We use this to study the relation between the eta invariant sliceness obstruction, eta-invariant ribbonness obstruction, and the L 2 –eta invariant sliceness obstruction recently introduced by Cochran, Orr and Teichner. In particular we give an example of a knot which has zero eta invariant and zero metabelian L 2 –eta invariant sliceness obstruction but is not ribbon. 1.

Summary. 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. Key words: K- and L-groups of group rings and group C ∗-algebras, Baum-Connes

Let G be a torsion free discrete group and let Q denote the field of algebraic numbers in C. We prove that QG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups which are residually torsion free elementary amenable or which are residually free. This result implies that there are no non-trivial zero-divisors in CG. The statement relies on new approximation results for L 2-Betti numbers over QG, which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial Laplacian on L 2-cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers, whenever the covering transformation group is either amenable or in the Linnell class C. We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class G. MSC: 55N25 (homology with local coefficients), 16S34 (group rings, Laurent rings), 46L50 (non-commutative measure theory)

Abstract. It is shown in this paper that the topological phenomenon ”zero in the continuous spectrum”, discovered by S.P.Novikov and M.A.Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in certain abelian category. This approach implies homotopy invariance of the Novikov-Shubin invariants. Its main advantage is that it allows to use the standard homological techniques, such as spectral sequences, derived functors, universal coefficients etc., while studying the Novikov-Shubin invariants. It also leads to some new quantitative invariants, measuring the Novikov-Shubin phenomenon in a different way, which are used in the present paper in order to strengthen the Morse type inequalities of S.P. Novikov and M.A. Shubin [NS1]. §0.

Abstract. In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, proving a conjecture in [DM]. We also prove that an arbitrary amenable covering space of a finite simplicial complex is of determinant class.

Abstract. We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in [CHMM] to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.