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We introduce and study a strong "thin triangle"' condition for directedgraphs, which generalises the usual notion of hyperbolicity for a metric space.We prove that finitely generated left cancellative monoids whose right Cayleygraphs satisfy this condition must be finitely presented with polynomial Dehnfunctions, and hence word problems in NP. Under the additional assumption ofright cancellativity (or in some cases the weaker condition of boundedindegree), they also admit algorithms for more fundamentallysemigroup-theoretic decision problems such as Green's relations L, R, J, D andthe 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 Cayleygraph is a essentially a tree (hence hyperbolic in our sense and probably anyreasonable sense), but which is not even recursively presentable. This seems tobe strong evidence that no geometric notion of hyperbolicity will be strongenough to yield much information about finitely generated monoids in absolutegenerality.

A strong geometric hyperbolicity property for directed graphs and monoids