Spaces of surface group representations

Abstract

Let $\Gamma _g$ denote the fundamental group of a closed surface of genus $g \ge 2$. We show that every geometric representation of $\Gamma _g$ into the group of orientation-preserving homeomorphisms of the circle is rigid, meaning that its deformations form a single semi-conjugacy class. As a consequence, we give a new lower bound on the number of topological components of the space of representations of $\Gamma _g$ into ${{\mathrm{Homeo}}}_+(S^1)$. Precisely, for each nontrivial divisor $k$ of $2g-2$, there are at least $|k|^{2g} + 1$ components containing representations with Euler number $\frac{2g-2}{k}$. Our methods apply to representations of surface groups into finite covers of ${{\mathrm{PSL}}}(2,\mathbb {R})$ and into ${{\mathrm{Diff}}}_+(S^1)$ as well, in which case we recover theorems of W. Goldman and J. Bowden. The key technique is an investigation of stability phenomena for rotation numbers of products of circle homeomorphisms using techniques of Calegari–Walker. This is a new approach to studying deformation classes of group actions on the circle, and may be of independent interest.