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Abstract: Main theorems of the article concern the problem of M. Atiyah on possiblevalues of l^2-Betti numbers. It is shown that all non-negative real numbers arel^2-Betti numbers, and that -many- for example all non-negative algebraicreal numbers are l^2-Betti numbers of simply connected manifolds with respectto a free cocompact action. Also an explicit example is constructed which leadsto a simply connected manifold with a transcendental l^2-Betti number withrespect to an action of the threefold direct product of the lamplighter groupZ-2 wr Z. The main new idea is embedding Turing machines into integral grouprings. The main tool developed generalizes known techniques of spectralcomputations for certain random walk operators to arbitrary operators ingroupoid rings of discrete measured groupoids.



Autor: Łukasz Grabowski

Fuente: https://arxiv.org/







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