Warren Dicks and Peter A. Linnell,
L2-Betti numbers of one-relator groups,
Math. Ann., 337 (2007), 855-874.       

Abstract:  Surface-plus-one-relation groups were introduced by Hempel who called them one-relator
surface groups
.  
   We determine the L2-Betti numbers of all one-relator groups and all surface-plus-one-relation groups.  
In particular, we show that for any such group G, the L2-Betti numbers bn(2)(G) vanish for all n>1.
    We also obtain some information about the L2-cohomology of left-orderable groups, and deduce
the non-L2  result that, in any left-orderable group of homological dimension one, all two-generator
subgroups are free.

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