Zariski-Closures of Linear Reflection Groups

Abstract

We give necessary and sufficient conditions for a linear reflection group in the sense of Vinberg to be Zariski-dense in the ambient projective general linear group. As an application, we show that every irreducible right-angled Coxeter group of rank $N\ge 3$ virtually embeds Zariski-densely in ${\mathrm{SL}}_n(\mathbb{Z})$ for all $n\ge N$. This allows us to settle the existence of Zariski-dense surface subgroups of ${\mathrm{SL}}_n(\mathbb{Z})$ for all $n\ge 3$. Among the other applications are examples of Zariski-dense one-ended finitely generated subgroups of ${\mathrm{SL}}_n(\mathbb{Z})$ that are not finitely presented for all $n\ge 6$.

34 pages, 4 figures. Comments welcome