2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130856We 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.18 pages, version 3, minor changes. To appear in Math. AnnGroup TheoryGeometric Topology20F05, (Primary) 16S34, 20J05 (Secondary)text