Groups of small homological dimension and the Atiyah Conjecture

2004/01/23 by Peter Kropholler, Peter H. Kropholler, Peter A. Linnell +5
Mathematics · #18G20 #46L99 #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #msc:18G20 #msc:46L99

paper · pdf · doi:10.48550/arxiv.math/0401312

5 pages

arxiv created 2004/01/23 · openalex publication_date 2004/01/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Dimension theory of arbitrary modules over finite von Neumann algebras and applications to L²-Betti numbers
• Dimension of elementary amenable groups
• Graphs of groups and the Atiyah conjecture for one-relator groups
• L²-Betti Numbers of Locally Compact Groups
• Homological Dimension of Solvable Groups