  In this article, we study geometric properties of nilpotent groups. We find a
geometric criterion for the word problem for the finitely generated free
nilpotent groups. By geometric criterion, we mean a way to determine whether
two words represent the same element in a free nilpotent group of rank $r$ and
class $k$ by analyzing their behavior on the Cayley graph of the free nilpotent
group of rank $r$ and class $k-1$.
