Hyperlinearity, essentially free actions and \(L^2\)-invariants. The sofic property. (English) Zbl 1070.43002
The notion of sofic group was introduced by Gromov in the year 1999. In this paper the authors give a new characterization of sofic groups and prove the following three major results:
1) All countable sofic groups are hyperlinear, thereby establishing the validity of the Connes embedding conjecture for sofic groups.
2) The determinant conjecture holds for sofic groups.
3) The Atiyah conjecture holds for the limit group of a convergent sequence \((\Gamma_n, S_n)_{n\geq 1}\) of torsion free sofic groups for which the Atiyah conjecture holds.
1) All countable sofic groups are hyperlinear, thereby establishing the validity of the Connes embedding conjecture for sofic groups.
2) The determinant conjecture holds for sofic groups.
3) The Atiyah conjecture holds for the limit group of a convergent sequence \((\Gamma_n, S_n)_{n\geq 1}\) of torsion free sofic groups for which the Atiyah conjecture holds.
Reviewer: Madathum K. Viswanath (Chennai)
MSC:
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 55N25 | Homology with local coefficients, equivariant cohomology |
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