Arc coordinates for maximal representations

TL;DR

This paper extends arc coordinate methods to maximal representations of hyperbolic surfaces into PSp(4,R), focusing on pairs of pants, and introduces geometric parameters for visualization and parametrization of these representations.

Contribution

It generalizes arc coordinates for maximal representations into PSp(4,R) and introduces geometric parameters for visualization and parametrization of these representations.

Findings

Parametrization of maximal representations of the reflection group W_3 into PSp(4,R)

Introduction of geometric parameters within right-angled hexagons in Siegel space

Visualization of hexagons as polygonal chains in hyperbolic plane

Abstract

We generalize arc coordinates for maximal representations from a hyperbolic surface with boundary into $\text{PSp}(4,\mathbb{R})$, focusing on the case where the surface is a pair of pants. We introduce geometric parameters within the space of right-angled hexagons in the Siegel space $\mathcal{X}$. These parameters enable the visualization of a right-angled hexagon as a polygonal chain inside the hyperbolic plane $\mathbb{H}^{2}$. We explore the geometric properties of reflections in $\mathcal{X}$ and introduce the notion of maximal representation of the reflection group $W_{3}=\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}$. We parametrize maximal representations from $W_{3}$ into $\text{PSp}^{\pm}(4,\mathbb{R})$, this induces a natural parametrization of a subset of maximal and Shilov hyperbolic representations into $\text{PSp}(4,\mathbb{R})$.