Divergence of finitely presented subgroups of CAT(0) groups

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 $\alpha \in [2,\mathrm{\infty })$ and $r^q\log (r)$ for integers $q\ge 2$. The same construction also yields examples of $CAT(0)$ groups containing contracting elements which are not contracting in certain finitely presented subgroups.