  We study a class of two-generator two-relator groups, denoted $J_n(m,k)$,
that arise in the study of relative asphericity as groups satisfying a
transitional curvature condition. Particular instances of these groups occur in
the literature as finite groups of intriguing orders. Here we find infinite
families of non-elementary virtually free groups and of finite metabelian
non-nilpotent groups, for which we determine the orders. All Mersenne primes
arise as factors of the orders of the non-metacyclic groups in the class, as do
all primes from other conjecturally infinite families of primes. We classify
the finite groups up to isomorphism and show that our class overlaps and
extends a class of groups $F^{a,b,c}$ with trivalent Cayley graphs that was
introduced by C.M.Campbell, H.S.M.Coxeter, and E.F.Robertson. The theory of
cyclically presented groups informs our methods and we extend part of this
theory (namely, on connections with polynomial resultants) to "bicyclically
presented groups" that arise naturally in our analysis. As a corollary to our
main results we obtain new infinite families of finite metacyclic generalized
Fibonacci groups.
