118 research outputs found
A spectral sequence to compute L2-Betti numbers of groups and groupoids
We construct a spectral sequence for L2-type cohomology groups of discrete
measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer
conjecture for aspherical manifolds with poly-surface fundamental groups. More
generally, we obtain a permanence result for the Hopf-Singer conjecture under
taking fiber bundles whose base space is an aspherical manifold with
poly-surface fundamental group. As further sample applications of the spectral
sequence, we obtain new vanishing theorems and explicit computations of
L2-Betti numbers of groups and manifolds and obstructions to the existence of
normal subrelations in measured equivalence relations.Comment: added remark 4.9 about applying spectral sequence in a non-ergodic
situation; minor correction
On Turing dynamical systems and the Atiyah problem
Main theorems of the article concern the problem of M. Atiyah on possible
values of l^2-Betti numbers. It is shown that all non-negative real numbers are
l^2-Betti numbers, and that "many" (for example all non-negative algebraic)
real numbers are l^2-Betti numbers of simply connected manifolds with respect
to a free cocompact action. Also an explicit example is constructed which leads
to a simply connected manifold with a transcendental l^2-Betti number with
respect to an action of the threefold direct product of the lamplighter group
Z/2 wr Z. The main new idea is embedding Turing machines into integral group
rings. The main tool developed generalizes known techniques of spectral
computations for certain random walk operators to arbitrary operators in
groupoid rings of discrete measured groupoids.Comment: 35 pages; essentially identical to the published versio
Orbit Equivalence and Measured Group Theory
We give a survey of various recent developments in orbit equivalence and
measured group theory. This subject aims at studying infinite countable groups
through their measure preserving actions.Comment: 2010 Hyderabad ICM proceeding; Dans Proceedings of the International
Congress of Mathematicians, Hyderabad, India - International Congress of
Mathematicians (ICM), Hyderabad : India (2010
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