Abstract: The notion of hyperbolicity is very significant in groups. Extending the notion to semigroup is interesting and challenging. Capturing a geometric property in terms of formal language theory has been proven to be very successful in the context of automatic groups and semigroups. Robert Gilman and Andrew Duncan used a push down automata to give an elegant 'formal language' definition of hyperbolic groups which naturally extends to semigroups.
This new class of semigroups seams to not share the computational properties of hyperbolic groups.
In this talk we will look at the different classes stemming from Gilman and Duncans idea, their relationship to other classes of semigroups with a special emphasis of computational properties.