  We study the relative growth of finitely generated subgroups in finitely
generated groups, and the corresponding distortion function of the embeddings.
We explore which functions are equivalent to the relative growth functions and
distortion functions of finitely generated subgroups. We also study the
connections between these two asymptotic invariants of group embeddings. We
give conditions under which a length function on a finitely generated group can
be extended to a length function on a larger group.
