(preprint, 2013) (with A. Valette) L^2-Betti numbers and Plancherel measure
We show how to compute the L^2-Betti numbers of type 1 groups in terms of the Plancherel measure and (ordinary) dimension of (reduced) cohomology with coefficients in irreducible representations. This in particular gives a very direct way to compute the L^2-Betti numbers for lattices in e.g. semisimple Lie groups and algebraic groups, as well as groups of automorphisms of trees. Combining with the results from the cross sections paper with David and Stefaan (see below), one even gets some cute consequences for the cohomologies with coefficients in irredicuble representaitons.
(preprint, 2013) (with D. Kyed and S. Vaes) L^2-Betti numbers of locally compact groups and their cross section equivalence relations
In this paper we prove the measure equivalence invariance of L^2-Betti numbers of locally compact unimodular groups using cocompact cross section equivalence relations. These in effect act as cocompact lattices in the group G in some measurable sense. This also allows improvement of several results from my thesis (see above), notably we show that the L^2-Betti numbers scale appropriately when passing to any unimodular closed subgroup with finite covolume, and that one can compute the L^2-Betti numbers of any unimodular locally compact groups equivalently as the von Neumann dimension of the reduced and the unreduced cohomology spaces. the techniques then also allow a direct transfer of several known results for countable equivalnce relations, in particular the vanishing of reduced L^2 cohomology for amenable groups is deduced in this way.
(Thesis, 2012)L^2-Betti Numbers of Locally Compact Groups
One motivation for this work is to unify definitions of the L^2-Betti numbers for (countable) discrete groups and (countable) locally finite graphs, the latter introduced by D. Gaboriau in this paper. This is done by giving a general definition of L^2-Betti numbers for locally compact unimodular groups in terms of the continuous cohomology, analogously to Lueck's definition for discrete groups. A key feature of L^2-Betti numbers compared to just the classical Betti numbers is that they scale proportionally when passing to finite index subgroups. We also explore extensions of this phenomenon in the setting of passing to lattices and prove proportionality results for totally disconnected groups, and for groups admitting a cocompact lattice. The vanishing theorem of reduced L^2-cohomology of discrete groups due to Cheeger and Gromov is also extended to unimodular locally compact groups.
(preprint, 2012) (with N. Monod) An Obstruction to $\ell^p$-dimension
One could hope to extend von Neumann dimension from the Hilbert space setting to a more general $\ell^p$ setting, defining a dimension function e.g. for isometric representations of a countable discrete group on p-spaces. Concrete definitions have been given by Gournay and Hayes for amenable, resp. sofic groups. While the results of Gournay and Hayes are mostly affirmative, in this paper we show that, for p>2 any such dimension function will lack some basic properties of von Neumann dimension.
(To appear in Osaka J. Math.) (with D. Kyed) A groupoid approach to Lueck's amenability conjecture
Lueck's amenability conjecture states that a group G is amenable if and only if the inclusion of the group ring in the group von Neumann algebra has a certain flatness property (it is "dimension flat"). Lueck proved the "only if" part in his work on L^2-Betti numbers of countable discrete groups, and furthermore pointed out that if G contains a copy of the free group on two generators then this inclusion of rings cannot be dimension flat. The idea for this paper was to use Gaboriau-Lyons' measure-theoretic converse to the von Neumann problem to produce, for any given non-amenable G, a finite cyclic group A such that the wreath product of A and G is not "dimension flat" hence proving Lueck's conjecture "up to taking wreath products." There seem to be some difficulties in tracking down the exact group algebra inside the von Neumann algebra, but we do end up with two partial results: One is a characterization of amenability of G in terms of "dimension flatness" of groupoid algebras naturally associated with G. The other finds subalgebras that are not dimension flat and have countable linear dimension.
(IEE Transactions on Information Theory, 58(12), pp. 7021--7035, (2012)) (with F. Topsøe) Computation of universal objects for distributions over co-trees (published .pdf) (Direct .pdf)
In this paper, we compute in closed form the universal code (a.k.a. universal predictor) for the model consisting of distributions respecting the order structure given by an arbitrary (finite) co-tree.
We also develop an algorithm to compute it exactly, in polynomial time.
(preprint) Composability in a certain family of entropies
I give a proof that in a somewhat general family of entropies, the Tsallis entropies are characterized as the ones enjoying the property of composability.
This was apparently known to the physicists, though I never did manage to understand how. In any case, the proof relies on no physical assumptions.