Connected components of the space of type-preserving representations

TL;DR

This paper fully characterizes the connected components of the space of type-preserving representations of punctured surface groups into PSL(2,R), revealing their indexing by Euler classes and peripheral element signs, with special cases for punctured spheres.

Contribution

It completes the classification of connected components for these representation spaces, including new components for punctured spheres with non-hyperbolic representations.

Findings

Connected components indexed by Euler classes and peripheral signs.

Additional components for punctured spheres with non-hyperbolic representations.

Complete count of all connected components in the space.

Abstract

We complete the characterization of the connected components of the space of type-preserving representations of a punctured surface group into $\mathrm{PSL}(2,\mathbb{R})$. We show that the connected components are indexed by the relative Euler classes and the signs of the images of the peripheral elements satisfying a generalized Milnor-Wood inequality; and when the surface is a punctured sphere, there are additional connected components consisting of ``totally non-hyperbolic" representations. As a consequence, we count the total number of the connected components of the space of type-preserving representations.