Components of spaces of surface group representations

TL;DR

This paper establishes new lower bounds on the number of connected components in the space of surface group representations into circle homeomorphisms, revealing rigidity phenomena and introducing a novel approach based on rotation number maximality.

Contribution

It provides the first lower bounds on the number of connected components for these representation spaces and introduces a new method using rotation number local maximality phenomena.

Findings

At least k^(2g) + 1 components with specific Euler numbers

Identification of rigid representations with fixed deformation classes

Application of new techniques to representations into finite covers of PSL(2,R) and Diff+(S^1)

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.