Schottky groups and maximal representations

TL;DR

This paper constructs Schottky type subgroups in automorphism groups of cyclically ordered sets, applies this to Hermitian symmetric spaces, and links them to maximal surface group representations, providing explicit fundamental domains.

Contribution

It introduces a new construction of Schottky subgroups in automorphism groups and connects them to maximal surface group representations in Hermitian symmetric spaces.

Findings

Schottky subgroups correspond to maximal representations in this setting

Explicit fundamental domains for maximal representations into Sp(2n,R) are constructed

The construction applies to the Shilov boundary of Hermitian symmetric spaces

Abstract

We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces with boundary. As an application, we construct explicit fundamental domains for the action of maximal representations into $\mathrm{Sp}(2n,\mathbb{R})$ on $\mathbb{RP}^{2n-1}$.