  We examine distortion of finitely generated normal subgroups. We show a
connection between subgroup distortion and group divergence. We suggest a
method computing the distortion of normal subgroups by decomposing the whole
group into smaller subgroups. We apply our work to compute the distortion of
normal subgroups of graph of groups and normal subgroups of right-angled Artin
groups that induce infinite cyclic quotient groups. We construct normal
subgroups of $\CAT(0)$ groups introduced by Macura and introduce a collection
of normal subgroups of right-angled Artin groups. These groups provide a rich
source to study the connection between subgroup distortion and group divergence
on $\CAT(0)$ groups.
