L^2-Betti numbers of one-relator groups
Group Theory
2007-06-13 v3 Geometric Topology
Abstract
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.
Cite
@article{arxiv.math/0508370,
title = {L^2-Betti numbers of one-relator groups},
author = {Warren Dicks and Peter A. Linnell},
journal= {arXiv preprint arXiv:math/0508370},
year = {2007}
}
Comments
18 pages, version 3, minor changes. To appear in Math. Ann