Flexibility of some subgroups of $Homeo_+(\mathbb{R})$

Juan Alonso, Joaquin Brum, Cristóbal Rivas

Published 2016-05-24Version 1

We study the space of representations $Hom(\pi_1(\Sigma), Homeo_+(\mathbb{R}))$, where $\pi_1(\Sigma)$ is the fundamental group of a hyperbolic closed surface, and $Homeo_+(\mathbb{R})$ is the group of order-preserving homeomorphisms of the real line. We focus on the subspace $Hom_{\#}$ of the representations with no global fixed points. We show that any such representation is {\em flexible}, meaning that its semi-conjugacy class has empty interior in $Hom_{\#}$. We also prove that $Hom_{\#}$ is connected and dense in $Hom(\pi_1(\Sigma),Homeo_+(\mathbb{R}))$. Our technique is based on the study of the varieties ${\mathcal V}_h$ of pairs of homeomorphisms of the line $(f,g)$ satisfying $[f,g]=h$ for a given $h$. We obtain a form of flexibility on ${\mathcal V}_h$ and semicontinuity of the map $h\mapsto {\mathcal V}_h$ with respect to some perturbations of $h$.