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Zariski-Closures of Linear Reflection Groups

Jacques Audibert, Sami Douba, Gye-Seon Lee, Ludovic Marquis

TL;DR

This work identifies precise algebraic conditions under which a Vinberg-type linear reflection group is Zariski-dense in the ambient linear group, showing that a non-symmetrizable Cartan matrix forces the Zariski-closure to be the full $ ext{SL}^{ ext{±}}(V)$, while preserving a nondegenerate symmetric form yields an orthogonal-type closure. Leveraging this, the authors construct extensive thin subgroups in $ ext{SL}_n( ext{Z})$, including Zariski-dense embeddings of irreducible right-angled Coxeter groups for all $n eq N$ (and $n\

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 $N \geq 3$ virtually embeds Zariski-densely in $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq N$. This allows us to settle the existence of Zariski-dense surface subgroups of $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq 3$. 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 $n \geq 6$.

Zariski-Closures of Linear Reflection Groups

TL;DR

This work identifies precise algebraic conditions under which a Vinberg-type linear reflection group is Zariski-dense in the ambient linear group, showing that a non-symmetrizable Cartan matrix forces the Zariski-closure to be the full , while preserving a nondegenerate symmetric form yields an orthogonal-type closure. Leveraging this, the authors construct extensive thin subgroups in , including Zariski-dense embeddings of irreducible right-angled Coxeter groups for all (and $n\

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 virtually embeds Zariski-densely in for all . This allows us to settle the existence of Zariski-dense surface subgroups of for all . Among the other applications are examples of Zariski-dense one-ended finitely generated subgroups of that are not finitely presented for all .

Paper Structure

This paper contains 24 sections, 35 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Applications to thin subgroups of $\mathrm{SL}_n(\mathbb{Z})$
  3. Zariski-dense witnesses to incoherence of $\mathrm{SL}_n(\mathbb{Z})$
  4. Relation to previous work of Benoist
  5. Vinberg's theory of reflection groups
  6. Coxeter groups
  7. Cartan matrices
  8. Reflection groups
  9. Reducing to an irreducible representation
  10. Large reflection groups always contain a matrix $D (\lambda)$
  11. Large Coxeter groups contain standard quasi-Lannér Coxeter groups
  12. Proof of Proposition \ref{['prop_dlambda']} in the presence of an invariant round domain
  13. Proof of Proposition \ref{['prop_dlambda']} in the general case
  14. Lie groups containing a matrix $D(\lambda)$
  15. Proof of Theorem \ref{['thm:coxeter']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $W$ be a finitely generated Coxeter group that is not virtually abelian and $\rho: W \rightarrow \mathrm{GL}(V)$ a representation of $W$ as a reflection group (see Def. def:refl_gp). Suppose that $\rho$ is irreducible.

Figures (4)

  • Figure 1: A family of compact hyperbolic Coxeter $3$-polytopes $P_k$
  • Figure 2: The Coxeter diagrams of the hyperbolic $3$-polytope $P_1$ and the hyperbolic $4$-polytope $Q_1$
  • Figure 3: A family of Coxeter groups
  • Figure 4: A Coxeter group $W_m$ as in the proof of Theorem \ref{['thm:incoherence']}

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • ...and 64 more