L^2-Betti numbers of one-relator groups

Details L^2-Betti numbers of one-relator groups L^2-Betti numbers of one-relator groups

2005-08-19

arxiv.org/abs/math/0508370v3

Math. Ann. 337 (2007), no. 4, 855-874

Mathematics

Group Theory

18 pages, version 3, minor changes. To appear in Math. Ann

Scientific paper

10.1007/s00208-006-0058-y

We determine the L^2-Betti numbers of all one-relator groups and all surface-plus-one-relation groups (surface-plus-one-relation groups were introduced by Hempel who called them one-relator surface groups). In particular we show that for all such groups G, the L^2-Betti numbers b_n^{(2)}(G) are 0 for all n>1. We also obtain some information about the L^2-cohomology of left-orderable groups, and deduce the non-L^2 result that, in any left-orderable group of homological dimension one, all two-generator subgroups are free.

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