The definability criteria for convex projective polyhedral reflection groups

Following Vinberg, we find the criteria for a subgroup generated by reflections Γ ⊂ SL ± ( n + 1 , R ) and its finite-index subgroups to be definable over A where A is an integrally closed Noetherian ring in the field R . We apply these criteria 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 ± ( n + 1 , Z ) . Finally we provide some examples of SL ± ( n + 1 , 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 ± ( n + 1 , R ) that are definable over Z . These were known by Goldman, Benoist, and so on previously.