Schottky presentations of positive representations

TL;DR

This paper establishes a connection between positive representations and Schottky groups via a new partial cyclic order on flag manifolds, providing geometric constructions of fundamental domains.

Contribution

It introduces a partial cyclic order based on 3-hyperconvexity, linking positive representations to Schottky groups and constructing explicit fundamental domains.

Findings

Schottky groups correspond to positive representations

A new partial cyclic order on flag manifolds is defined

Explicit polyhedral fundamental domains are constructed

Abstract

We show that the notion of $3$-hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct polyhedral fundamental domains for the domain of discontinuity that these groups admit in the projective space or the sphere, depending on the dimension.