Orderings and flexibility of some subgroups of Homeo(+)(R)

Abstract

In this work we exhibit flexibility phenomena for some (countable) groups acting by order preserving homeomorphisms of the line. More precisely, we show that if a left orderable group admits an amalgam decomposition of the form G = F-n *(Z) F-m where n + m >= 3, then every faithful action of G on the line by order preserving homeomorphisms can be approximated by another action (without global fixed points) that is not semi-conjugated to the initial action. We deduce that LO(G), the space of left orders of G, is a Cantor set. In the special case where G = pi(1)(Sigma) is the fundamental group of a closed hyperbolic surface, we found finer techniques of perturbation. For instance, we exhibit a single representation whose conjugacy class in dense in the space of representations. This entails that the space of representations without global fixed points of pi(1)(Sigma) into Homeo(+)(R) is connected, and also that the natural conjugation action of pi 1(Sigma) on LO(pi(1)(Sigma)) has a dense orbit.