Consider a group generated by a geometrically fast collection of one-orbital increasing homeomorphisms (i.e. bumps). It is shown by work of Bleak et al. that if two such collections have isomorphic ascending sequences of transition points then they generate isomorphic groups. Replacing bumps with conjugates by other bumps witnesses that many collections with non-isomorphic sequences will generate groups belonging to a single isomorphism class.
As such, it is an open question whether any dynamical diagram of n bumps where none are isolated generates $F_n$. In the case that the collection consists of four homeomorphisms, this reduces to: is $PF_4$ isomorphic to $F_4$? In this talk we will answer this question in the affirmative by showing that groups generated by geometrically fast collections of bumps are diagram groups. Joint work with Jim Belk.