  We construct families of $CAT(0)$ groups containing finitely presented groups
whose divergence functions are of the form $r^\alpha$ for a dense set of
exponents $\alpha \in [2,\infty)$ and $r^q\log(r)$ for integers $q \geq 2$. The
same construction also yields examples of $CAT(0)$ groups containing
contracting elements which are not contracting in certain finitely presented
subgroups.
