The definability criterions for convex projective polyhedral reflection groups

TL;DR

This paper establishes criteria for when reflection groups acting on convex projective domains are definable over certain rings, and applies these to construct and classify specific Coxeter groups and their representations.

Contribution

It provides new definability criteria for convex projective reflection groups and develops methods to construct and classify their representations over integers.

Findings

Criteria for definability over rings for reflection groups

Construction of injective homomorphisms into SL(n+1,Z)

Classification of conjugacy classes of reflection subgroups over Z

Abstract

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.