Irreducible fast sets of bump homeomorphisms generate copies of Thompson's groups $F_n$

Abstract

A homeomorphism of an interval is a positive bump if its support is a single open interval on which it moves every point to the right. Choosing a fundamental domain $[m,b(m))$ for the action of a positive bump $b$ on its support splits the remainder of the support into two intervals, called the feet of $b$. A finite set of positive bumps is geometrically fast if fundamental domains can be chosen so that all the resulting feet are pairwise disjoint. The crossing graph of such a set has the bumps as its vertices, two bumps being adjacent whenever their supports overlap but are not nested, and the set is irreducible if its crossing graph is connected. We prove that for every $n\geq 2$, every group generated by an irreducible geometrically fast set of $n$ positive bumps is isomorphic to the $n$-ary Thompson group $F_n$. This answers the strong version of a problem posed by Brin and Zaremsky (Oberwolfach Rep. 15 (2018)).