Following Vinberg, we find the criterions for a subgroup generated by reflections $\Gamma \subset \SL^{ \pm}(n+1,\mathbb{ R})$ and its finite-index subgroups to be definable over $\mathbb{ A}$ where $\mathbb{ A}$ is an integrally closed Noetherian ring in the field $\mathbb{ R}$. We apply the criterions for groups generated by reflections that act cocompactly on irreducible properly convex open subdomains of the $n$-dimensional projective sphere. This gives a method for constructing 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 $\mathbb{ Z}$. These were known by Goldman, Benoist, and so on previously.
Categories: math.GT