Brin and Sapir have conjectured that every subgroup of Richard Thompson's group F is either elementarily amenable or else contains a copy of F. In this talk I will isolate the notion of a fast generating set for a finitely generated subgroups of the homeomorphism group of the interval. On one hand, groups with fast generating sets admit a nice theory (for instance they are all embeddable into F and are each characterized by the dynamics of their generators). On the other hand they seem to represent a fragment of the Brin-Sapir conjecture which is at the same time substantial and more tractable. This talk will give an analysis of the Brin-Sapir conjecture and will report on joint work in progress with Collin Bleak, Matt Brin, and Martin Kassabov.