Concise presentations of direct products

Abstract

Direct powers of perfect groups admit more concise presentations than one might naively suppose. If H 1 ( G , Z ) = H 2 ( G , Z ) = 0 H_1(G,\mathbb {Z})=H_2(G,\mathbb {Z})=0 , then G n G^n has a presentation with O ( log ⁡ n ) O(\log n) generators and O ( log ⁡ n ) 3 O(\log n)^3 relators. If, in addition, there is an element g ∈ G g\in G that has infinite order in every non-trivial quotient of G G , then G n G^n has a presentation with d ( G ) + 1 d(G) +1 generators and O ( log ⁡ n ) O(\log n) relators. The bounds that we obtain on the deficiency of G n G^n are not monotone in n n ; this points to potential counterexamples for the Relation Gap Problem.