Finiteness of integral representations on 2-perfect truncation polytopes

TL;DR

This paper proves that the deformation space of properly convex real projective structures on certain hyperbolic Coxeter orbifolds contains only finitely many integral representations, extending to a broader class of polytopes.

Contribution

It establishes finiteness of integral representations in the deformation space for hyperbolic Coxeter truncation polytopes and related classes.

Findings

Finiteness of integral representations in the deformation space for specific Coxeter orbifolds.

Extension of finiteness result to irreducible, large, 2-perfect truncation polytopes.

Abstract

Let $P$ be a compact hyperbolic Coxeter truncation polytope of dimension $d\ge 3$, and let $\Gamma$ be the orbifold fundamental group of the associated Coxeter orbifold $\mathcal{O}_P$. Let $\mathscr{G}(\Gamma,G)$ be the geometric component containing the holonomy representation in $\operatorname{Hom}(\Gamma,G)/G$. $\mathscr{G}(\Gamma,G)$ is identified with the deformation space of properly convex real projective structures on the Coxeter orbifold $\mathcal{O}_P$. We prove that $\mathscr{G}(\Gamma,G)$ contains only finitely many integral representations. The same conclusion holds more generally for irreducible, large, $2$-perfect truncation polytopes.