The Baum-Connes conjecture.
Operator -theory for groups which act properly and isometrically on Hilbert space.
The Novikov conjecture for linear groups
Let K be a field. We show that every countable subgroup of GL(,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(,K) is exact, in the sense of C-algebra theory.
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