The definability criterion for cocompact convex projective polyhedral reflection groups

Abstract

In this paper, we prove the criterion for a Zariski dense subgroup generated by reflections $\Gamma \subset \SL^{\pm}(n+1,\mathbb{R})$ to be definable over $\mathbb{A}$ where $\mathbb{A}$ is an integrally closed Noetherian ring in the field $\mathbb{R}$. We apply this criterion for groups generated by reflections that act cocompactly on irreducible properly convex open subdomains of the $n$-dimensional projective sphere. This gives a methodology to construct injective group homomorphisms from such Coxeter groups to $\SL^{\pm}(n+1,\mathbb{Z})$. Finally we provide some examples of $\SL^{\pm}(n+1,\mathbb{Z})$-representations of such Coxeter groups. In particular, we consider simplicial reflection groups that are isomorphic to hyperbolic simplicial groups and classify all the conjugacy classes of the reflection subgroups in $\SL^{\pm}(n+1,\mathbb{R})$ that are definable over $\mathbb{Z}$.