We introduce and study a strong "thin triangle"' condition for directed
graphs, which generalises the usual notion of hyperbolicity for a metric space.
We prove that finitely generated left cancellative monoids whose right Cayley
graphs satisfy this condition must be finitely presented with polynomial Dehn
functions, and hence word problems in NP. Under the additional assumption of
right cancellativity (or in some cases the weaker condition of bounded
indegree), they also admit algorithms for more fundamentally
semigroup-theoretic decision problems such as Green's relations L, R, J, D and
the corresponding pre-orders.
In contrast, we exhibit a right cancellative (but not left cancellative)
finitely generated monoid (in fact, an infinite class of them) whose Cayley
graph is a essentially a tree (hence hyperbolic in our sense and probably any
reasonable sense), but which is not even recursively presentable. This seems to
be strong evidence that no geometric notion of hyperbolicity will be strong
enough to yield much information about finitely generated monoids in absolute
generality.Comment: Exposition improved. Results unchange
