The Novikov conjecture for linear groups

Guentner, Erik; Higson, Nigel; Weinberger, Shmuel

Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 243-268.

Let $\mathrm{K}$ be a field. We show that every countable subgroup of $\mathrm{GL}(n,\mathrm{K})$ 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 $\mathrm{GL}(2,\mathrm{K})$ 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 $\mathrm{GL}(n,\mathrm{K})$ is exact, in the sense of $C^{*}$-algebra theory.

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