Components of spaces of surface group representations

Abstract

We give a new lower bound on the number of connected components of the space of representations of a surface group into the group of orientation preserving homeomorphisms of the circle. Precisely, for the fundamental group of a genus g surface, we show there are at least k^(2g) + 1 connected components containing representations with Euler number (2g-2)/(k). We also show that certain representations are rigid, meaning that all deformations lie in the same semiconjugacy class. Our methods apply to representations of surface groups into finite covers of PSL(2,R) and into Diff+(S^1) as well, in which case we recover theorems of W. Goldman and J. Bowden. The key technique is an investigation of local maximality 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.