Faithful semigroup diagram representations of homeomorphism groups

In this thesis we investigate two different classes of objects. The first is the class of Dénes cycles of a tree; we show a ‘strong’ correspondence between such cycles and certain partial orders on the edges of the tree. This allows us to find a slick new proof of a classical result, as well as give an algorithm for computing the multiplicity of a Dénes cycle. The second class of objects we investigate, and the main focus of this thesis, is groups generated by geometrically fast sets of bumps. We show that the class of such groups on the interval coincides with a certain subclass of diagram groups, and then exploit this connection to answer a question of Matthew Brin. We then go on to consider groups generated by fast bumps in a general setting and, in so doing, find a necessary and sufficient condition for when groups generated by fast sets of bumps on the circle are isomorphic to certain annular diagram groups. To finish, we deduce a couple of properties of fast groups of the interval using their diagram group representations. In aid of this, we define a group structure on infinite tree diagrams.