Finiteness of integral representations on 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.