Abstract: “A diagram is a geometric object associated with a derivation between two equivalent words in a semigroup presentation, and can be thought of as a two dimensional analogue of words on an alphabet. A diagram group is formed by all (reduced) diagrams associated with derivations from some word back to itself, for some semigroup presentation. These groups have deep connections with groups of increasing homeomorphisms and a number of familiar groups appear; most notably, Thompson’s group F is the diagram group of < x | xx=x >. Indeed, any diagram group can be represented as a group of increasing homeomorphisms, though not always faithfully. This raises the question: for which groups of increasing homeomorphisms does their exist a faithful representation as a diagram group? In this talk I show that groups generated by geometrically fast homeomorphisms with one orbital form a class of such groups and, in doing so, develop a procedure for finding presentations for these groups.”