  We introduce a quantitative characterization of subgroup alternatives modeled
on the Tits alternative in terms of group laws and investigate when this
property is preserved under extensions. We develop a framework that lets us
expand the classes of groups known to have locally uniform exponential growth
to include extensions of either word hyperbolic or right-angled Artin groups by
groups with locally uniform exponential growth. From this, we deduce that the
automorphism group of a torsion-free one-ended hyperbolic group has locally
uniform exponential growth. Our methods also demonstrate that automorphism
groups of torsion-free one-ended toral relatively hyperbolic groups and certain
right-angled Artin groups satisfy our quantitative subgroup alternative.
