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.
September 24, 2006 version, 18 pages, available as
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