Non-orientable surface-plus-one-relation groups

Recently Dicks–Linnell determined the L2-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 L2-Betti numbers of the non-orientable surface-plus-one-relation groups.