  We study the free metabelian group $M(2,n)$ of prime power exponent $n$ on
two generators by means of invariants $M(2,n)'\to \mathbb{Z}_n$ that we
construct from colorings of the squares in the integer grid $\mathbb{R} \times
\mathbb{Z} \cup \mathbb{Z} \times \mathbb{R}$. 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 $R(2,n)$.

Group Theory;Algebraic Topology;Combinatorics