Let $G$ be a finitely generated group (optionally torsion-free). Let $N$ be a submonoid of $G$ (that is, a subsemigroup with $1$).

A (cancellative) monoid/semigroup $S$ is *right reversible* if for all $a,b \in S$, it follows that $Sa \cap Sb \neq\emptyset$. This is the Ore condition for semigroups, and allows one to invert the elements of $S$ so that the set of products, $S\cdot S^{-1}$, is a group.

Suppose that $G$ is nilpotent. Does it follow that every submonoid $N$ is right reversible?

What are the *right* conditions on $G$ to guarantee this property?

Something like nilpotent by finite comes to mind, since this implies the group ring ${\bf Z} G$ is Goldie. Obviously, this is a strong condition on $G$, since it fails for large classes of groups, e.g., those containing a non-abelian free group, or probably, those for which the group ring is not Goldie.

The question arose from a study of random walks on discrete groups, related to the boundary of one of the usual boundaries. It is a little too complicated to explain the connections here.