Approximating nonabelian free groups by groups of homeomorphisms of the real line

We show that for a large class C of finitely generated groups of orientation preserving homeomorphisms of the real line, the following holds: Given a group G of rank k in C, there is a sequence of k-markings (G,S-n), n is an element of N whose limit in the space of marked groups is the free group of rank k with the standard marking. The class we consider consists of groups that admit actions satisfying mild dynamical conditions and a certain "self-similarity" type hypothesis. Examples include Thompson's group F, Higman-Thompson groups, Stein-Thompson groups, various Bieri-Strebel groups, the golden ratio Thompson group, and finitely presented nonamenable groups of piecewise projective homeomorphisms. For the case of Thompson's group F we provide a new and considerably simpler proof of this fact proved by Brin in [4]. (C) 2020 Elsevier Inc. All rights reserved.