Invariants for metabelian groups of prime power exponent, colorings, and stairs

Abstract We study the free metabelian group $M(2,n)$ of prime power exponent n on two generators by means invariants $M(2,n)'\to \mathbb {Z}_n$ that we construct from colorings squares in integer grid $\mathbb {R} \times {Z} \cup {R}$ . In particular, improve bounds found Newman for order $M(2,2^k)$ identities , which give information about Burnside $B(2,n)$ and restricted $R(2,n)$