David Kyed

L 2 -Betti numbers of locally compact groups and their cross section equivalence relations

We prove that the L-Betti numbers of a unimodular locally compact group G coincide, up to a natural scaling constant, with the L-Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of G. As a consequence, we obtain that the reduced and un-reduced L-Betti numbers of G agree and that the L-Betti numbers of a lattice Γ in G equal those of G up to scaling by the covolume of Γ in G. We also deduce several vanishing results, including the vanishing of the reduced L-cohomology for amenable locally compact groups.

Amenability and vanishing of L2-Betti numbers: An operator algebraic approach☆

Abstract We introduce a Folner condition for dense subalgebras in finite von Neumann algebras and prove that it implies dimension flatness of the inclusion in question. It is furthermore proved that the Folner condition naturally generalizes the existing notions of amenability and that the ambient von Neumann algebra of a Folner algebra is automatically injective. As an application, we show how our techniques unify previously known results concerning vanishing of L 2 -Betti numbers for amenable groups, quantum groups and groupoids and moreover provide a large class of new examples of algebras with vanishing L 2 -Betti numbers.

L-2-Betti Numbers Of Rigid C*-Tensor Categories And Discrete Quantum Groups

We compute the L-2-Betti numbers of the free C*-tensor categories, which are the representation categories of the universal unitary quantum groups A(u)(F). We show that the L-2-Betti numbers of the dual of a compact quantum group G are equal to the L-2-Betti numbers of the representation category Rep. (G) and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of L-2-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first L-2-Betti number in terms of a generating set of a C*-tensor category.

Higher

We show that the first \({\ell }^2\)-Betti number of the duals of the free unitary quantum groups is one, and that all \({\ell }^2\)-Betti numbers vanish for the duals of the quantum automorphism groups of full matrix algebras.

\ell^2

Using the Foelner condition for coamenable quantum groups we derive information about the ring theoretical structure of the Hopf algebras arising from such quantum groups, as well as an approximation result concerning the Murray von Neumann dimension associated with the corresponding the von Neumann algebra.

-Betti numbers of universal quantum groups

We prove that amenability of a discrete group is equivalent to dimension flatness of certain ring inclusions naturally associated with measure preserving actions of the group. This provides a group-measure space theoretic solution to a conjecture of Luck stating that amenability of a group is characterized by dimension flatness of the inclusion of its complex group algebra into the associated von Neumann algebra.