Concise presentations of direct products

TL;DR

This paper demonstrates that direct powers of certain perfect groups can be presented more concisely than expected, with bounds on generators and relators that depend logarithmically on the power, revealing new structural insights.

Contribution

It provides new bounds on the minimal presentations of direct powers of perfect groups, including cases with special elements, and discusses implications for the Relation Gap Problem.

Findings

Direct powers of perfect groups have O(log n) generators and relators.

Additional conditions lead to even more concise presentations with d(G)+1 generators.

Bounds on deficiency are non-monotonic, suggesting potential counterexamples for the Relation Gap Problem.

Abstract

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.