Complexity classes of groups of fast homeomorphisms of the interval

In this talk we discuss two flavours of results.  The first result gives a certain rigidity result applicable to a broad family of groups of homeomorphisms of the interval (the “Fast groups”), based on a broad generalisation of Fricke and Klein’s Ping-Pong Theorem.  The result gives a finite combinatorial characterisation of these groups, and a corollary of that result is that all fast groups embed in R. Thompson’s group $F$.  The second flavour of results describes an exploration of the complexity classes of the elementary amenable fast groups.  Namely, we demonstrate the existence of a family of fast, finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation in type $ε_0+1$.  All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. Joint with Brin, Kassabov, Moore, and Zaremsky.