  The Bishop-Gromov theorem is a comparison theorem of differential geometry
that upperbounds the growth of volume of a geodesic ball in a curved space. For
many spaces, this bound is far from tight. We identify a major reason the bound
fails to be tight: it neglects the effect of shear. By using higher curvature
invariants to lowerbound the average shear, we are able to place
tighter-than-Bishop-Gromov upperbounds on the late-time growth rates of
geodesic balls in homogeneous spaces with non-positive sectional curvature. We
also provide concrete guidance on how our theorem can be generalized to
inhomogeneous spaces, to spaces with positive sectional curvatures, and to
intermediate and short times. In arXiv:2209.09288 we discovered an enhancement
to the BG theorem that was strongest at early times, and that relied upon
additive properties of families of Jacobi equations; in this paper we find a
different enhancement at late times that connects to multiplicative properties
of families of Jacobi equations. A novel feature shared by both papers is the
consideration of families of equations that are not coupled but whose
coefficients are correlated.
