Thin right-angled Coxeter groups in some uniform arithmetic lattices

TL;DR

This paper demonstrates that certain right-angled Coxeter groups can be embedded as thin subgroups within uniform arithmetic lattices in indefinite orthogonal groups, revealing new connections between geometric group theory and arithmetic groups.

Contribution

It introduces a novel embedding technique for irreducible right-angled Coxeter groups into uniform arithmetic lattices in orthogonal groups.

Findings

Embedding of right-angled Coxeter groups into arithmetic lattices.

Extension of Agol's argument to new group embeddings.

Identification of conditions for thin subgroup embeddings.

Abstract

Using a variant of an unpublished argument due to Agol, we show that an irreducible right-angled Coxeter group on ${n \geq 3}$ vertices embeds as a thin subgroup of a uniform arithmetic lattice in an indefinite orthogonal group $\mathrm{O}(p, q)$ for some $p, q \geq 1$ satisfying $p+q=n$.