Mikhail Gromov introduced hyperbolic groups in 1987 through geometric properties of Cayley graphs, viewed as (geodesic) metric spaces. This class of groups captured very quickly the attention and the heart of (geometric)
group theorists due to a perfect combination of very high-valued properties:

* high computational potential (and many solvable properties);

* closed for quasi-isometry (a revolutionary closure operator in the context of group theory);

* a natural compactification, the space of ends (important for dynamics).

Another very nice class of groups, this one going a long way back, is the class of groups possessing good normal forms (which includes many non hyperbolic groups). Normal forms can be usually defined through rewriting systems with adequate properties. Rewriting systems can of course be used to define more general structures such as monoids.

However, (mathematical) common sense states that Cayley graphs of monoids cannot provide nice geometric properties because they are not strongly connected and one cannot build a distance without symmetry... and if common sense were wrong?

We shall present a class of rewriting systems, generating important classes of groups and other monoids, that produces good normal forms and has hyperbolic geometry. Moreover, the study of its space of ends can be applied to subjects such as endomorphism dynamics.