Fenchel-Nielsen Coordinates for Maximal Representations

TL;DR

This paper introduces Fenchel-Nielsen coordinates for maximal representations of surface groups into Sp(2n,R), extending classical concepts to higher rank groups and non-closed surfaces, with applications in counting components and analyzing limit curves.

Contribution

It develops a new coordinate system for maximal surface group representations into Sp(2n,R), including non-closed surfaces, and investigates their geometric and topological properties.

Findings

Counted the number of connected components of the representation space.

Proved continuity of the limit curve for certain representations.

Extended Fenchel-Nielsen coordinates to non-closed surfaces.

Abstract

We develop Fenchel-Nielsen coordinates for representations of surface groups into Sp(2n,R) with maximal Toledo invariant. Analogous to classical Fenchel-Nielsen coordinates on the Teichm\"uller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigation of the gluing. As applications we obtain results for non-closed surfaces, which have been known only for closed surfaces before: we count the number of connected components of their representation space and prove continuity for the limit curve for a certain type of representations.