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 of invariants M (2, n) ′ → Zn that we construct from colorings of the squares in the integer grid R × Z ∪ Z × R. In particular we improve bounds found by M.F. Newman for the order of M (2, 2 k ). We study identities in M (2, n), which give information about identities in the Burnside group B(2, n) and the restricted Burnside group R(2, n).