The Novikov conjecture for linear groups

Erik Guentner; Nigel Higson; Shmuel Weinberger

doi:10.1007/s10240-005-0030-5

The Novikov conjecture for linear groups

Erik Guentner
, Nigel Higson
, Shmuel Weinberger

Research output: Contribution to journal › Article › peer-review

86 Scopus citations

Abstract

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

Original language	English (US)
Pages (from-to)	243-268
Number of pages	26
Journal	Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques
Volume	101
Issue number	1
DOIs	https://doi.org/10.1007/s10240-005-0030-5
State	Published - Jun 2005

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s10240-005-0030-5

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N2 - Let K be a field. We show that every countable subgroup of GL(n,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(n,K) is exact, in the sense of C*-algebra theory.

AB - Let K be a field. We show that every countable subgroup of GL(n,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(n,K) is exact, in the sense of C*-algebra theory.

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The Novikov conjecture for linear groups

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