  We prove a general homological stability theorem for certain families of
groups equipped with product maps, followed by two theorems of a new kind that
give information about the last two homology groups outside the stable range.
(These last two unstable groups are the "edge" in our title.) Applying our
results to automorphism groups of free groups yields a new proof of homological
stability with an improved stable range, a description of the last unstable
group up to a single ambiguity, and a lower bound on the rank of the
penultimate unstable group. We give similar applications to the general linear
groups of the integers and of the field of order 2, this time recovering the
known stablility range. The results can also be applied to general linear
groups of arbitrary principal ideal domains, symmetric groups, and braid
groups. Our methods require us to use field coefficients throughout.
