2003/12/22 by Roman Sauer, Sauer, Roman · 2 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Geometric and Algebraic Topology #math.AT #math.GR #msc:37A20 #msc:46L85
paper · pdf · doi:10.48550/arxiv.math/0312411
17 pages
arxiv created 2003/12/22 · arxiv updated 2009/12/01
There are notions of L2-Betti numbers for discrete groups (Cheeger-Gromov, Lueck), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lueck's dimension theory,Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this article we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lueck's dimension theory, thereby encompassing the cases of groups,equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's,the L2-Betti numbers of a countable group G coincide with the L2-Betti numbers of the orbit equivalence relation of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.