Groups of small homological dimension and the Atiyah Conjecture
Peter Kropholler, Peter Linnell, Wolfgang Lueck
Abstract
A group G has homological dimension less or equal to 1 if it is locally free. We prove the converse provided that G satisfies the Atiyah Conjecture about L2-Betti numbers. We also show that a finitely generated elementary amenable group G of cohomological dimension less or equal to 2 possesses a finite 2-dimensional model for BG and in particular that G is finitely presented and the trivial ZG-module Z has a 2-dimensional resolution by finitely generated free ZG-modules.
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