Coxeter Groups are not higher rank Arithmetic Groups

TL;DR

This paper demonstrates that irreducible finitely generated Coxeter groups with certain geometric properties cannot contain finite index subgroups resembling higher rank arithmetic groups, highlighting limitations in their algebraic structure.

Contribution

It establishes a connection between the geometric representation of Coxeter groups and their inability to embed as higher rank lattices under specific conditions.

Findings

Coxeter groups with non-positive, non-degenerate Tits form lack higher rank lattice subgroups

The geometric representation constrains the algebraic structure of Coxeter groups

Certain Coxeter groups cannot be higher rank arithmetic groups

Abstract

Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits form of the Coxeter group is non-positive and non-degenerate, the Coxeter group does not contain any finite index subgroup isomorphic to an irreducible lattice in a semisimple group of R-rank greater or equal to 2.