Concise presentations of direct products

Direct powers of perfect groups admit more concise presentations than one might naively suppose. If H1(G;Z) = H2(G;Z) = 0, then Gn has a presentation with O(log n) generators and O(log n)3 relators. If, in addition, there is an element g 2 G that has infinite order in every non-trivial quotient of G, then Gn has a presentation with d(G) + 1 generators and O(log n) relators. The bounds that we obtain on the deficiency of Gn are not monotone in n; this points to potential counterexamples for the Relation Gap Problem.