## Abstract

We prove that the L
^{2}-Betti numbers of a unimodular locally compact group G coincide, up to a natural scaling constant, with the L
^{2}-Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of G. As a consequence, we obtain that the reduced and unreduced L
^{2}-Betti numbers of G agree and that the L
^{2}-Betti numbers of a lattice Γ in G equal those of G up to scaling by the covolume of Γ in G. We also deduce several vanishing results, including the vanishing of the reduced L
^{2}-cohomology for amenable locally compact groups.

Original language | English |
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Journal | Transactions of the American Mathematical Society |

Volume | 367 |

Issue number | 7 |

Pages (from-to) | 4917-4956 |

ISSN | 0002-9947 |

DOIs | |

Publication status | Published - 2015 |

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