  Recently Dicks-Linnell determined the $L^2$-Betti numbers of the orientable
surface-plus-one-relation groups, and their arguments involved some results
that were obtained topologically by Hempel and Howie. Using algebraic
arguments, we now extend all these results of Hempel and Howie to a larger
class of two-relator groups, and we then apply the extended results to
determine the $L^2$-Betti numbers of the non-orientable
surface-plus-one-relation groups.
