We generalize the concept of divergence of finitely generated groups by introducing
the upper and lower relative divergence of a finitely generated group with respect to
a subgroup. Upper relative divergence generalizes Gersten’s notion of divergence, and
lower relative divergence generalizes a definition of Cooper and Mihalik. While the
lower divergence of Alonso, Brady, Cooper, Ferlini, Lustig, Mihalik, Shapiro
and Short can only be linear or exponential, relative lower divergence can
be any polynomial or exponential function. In this paper, we examine the
relative divergence (both upper and lower) of a group with respect to a
normal subgroup or a cyclic subgroup. We also explore relative divergence of
groups and relatively hyperbolic groups with respect to various subgroups to better
understand geometric properties of these groups.
