Concise presentations of direct products

Direct powers of perfect groups admit more concise presentations than one might naively suppose. If $H_1G=H_2G=0$, then $G^n$ has a presentation with $O(\log n)$ generators and $O(\log n)^3$ relators. If, in addition, there is an element $g\in G$ that has infinite order in every non-trivial quotient of $G$, then $G^n$ has a presentation with $d(G) +1$ generators and $O(\log n)$ relators. The bounds that we obtain on the deficiency of $G^n$ are not monotone in $n$; this points to potential counterexamples for the Relation Gap Problem.
Comment: Final version. To appear in Proceedings of the American Mathematical Society. (9 pages, no figures)