Abstract. In this article we study cocycles of discrete countable groups with values in ℓ 2 G and the ring of affiliated operators U G. We clarify properties of the first cohomology of a group G with coefficients in ℓ 2 G and answer several questions from [14]. Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first ℓ 2 -Betti number. We give numerous applications and examples of groups which satisfy our assumptions.

Abstract. We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M∞-stable, homotopy-invariant, excisive Ktheory of algebras over a fixed unital ground ring H, (A, B) → kk * (A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel's homotopy algebraic K-theory, KH. We prove that, if H is commutative and A is central as an H-bimodule, then kk * (H, A) = KH * (A). We show further that some calculations from operator algebra KK-theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk.

We introduce a set of integrated interaction techniques to interpret and interrogate dimensionality-reduced data. Projection techniques generally aim to make a high-dimensional information space visible in form of a planar layout. However, the meaning of the resulting data projections can be hard to grasp. It is seldom clear why elements are placed far apart or close together and the inevitable approximation errors of any projection technique are not exposed to the viewer. Previous research on dimensionality reduction focuses on the efficient generation of data projections, interactive customisation of the model, and comparison of different projection techniques. There has been only little research on how the visualization resulting from data projection is interacted with. We contribute the concept of probing as an integrated approach to interpreting the meaning and quality of visualizations and propose a set of interactive methods to examine dimensionality-reduced data as well as the projection itself. The methods let viewers see approximation errors, question the positioning of elements, compare them to each other, and visualize the influence of data dimensions on the projection space. We created a web-based system implementing these methods, and report on findings from an evaluation with data analysts using the prototype to examine multidimensional datasets.

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the finite dimensional unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius normOur strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain non-residually finite central extensions of lattices in some simple p-adic Lie groups. These groups act on high rank Bruhat-Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov [9] have proved that any real zero polynomial in two variables has a determinantal representation. Brändén [2] has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. So the generalized Lax conjecture fails badly. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation, improving upon Brändén's mostly unconstructive result. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence.