  In this paper, we prove that infinite cancellative finitely generated
hyperbolic monoids never contain $\mathbb N\times\mathbb N$ as a submonoid but
that they contain an element of infinite order and, if they are elementary,
then they also contain a free monoid of rank at least 2. As a corollary we
obtain that the latter have exponential growth. We prove these results by
analysing the monoid of self-embeddings of hyperbolic digraphs and proving
fixed-point theorems for them.
