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The Novikov conjecture for linear groups. (English) Zbl 1073.19003
The main result of the paper is that every countable subgroup of \(GL(n,K)\) is uniformly embeddable in a Hilbert space, for any field \(K\). This implies split injectivity of the Baum-Connes assembly map for any coefficients, and the latter implies that the Novikov higher signature conjecture holds for these groups. It is also shown that the reduced group \(C^*\)-algebra of every subgroup of \(GL(n,K)\) is an exact \(C^*\)-algebra.
In the special case \(n=2\) the above result is strengthened: every countable subgroup of \(GL(2,K)\) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture with any coefficients holds for these groups.
The result on the Novikov conjecture is applied to the problem of homotopy invariance of relative eta invariants. Let \(M, M'\) be homotopy equivalent smooth, closed, oriented, odd-dimensional manifolds with fundamental group \(\pi\) and let \(\rho:\pi\to U(k)\) be a finite-dimensional unitary representation. It is shown that the difference \(\tilde{\eta}_\rho(M)-\tilde{\eta}_\rho(M')\) lies in the subring of \(\mathbb Q\) generated by \(\mathbb Z\), the inverses of the orders of torsion elements in \(\rho[\pi]\) and \(1/2\).

19K99 \(K\)-theory and operator algebras
46L85 Noncommutative topology
58J28 Eta-invariants, Chern-Simons invariants
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