Weyl groups, lattices and geometric manifolds

TL;DR

This paper explores the action of Weyl groups on root lattices to construct torsion-free subgroups in Coxeter groups, leading to new hyperbolic manifolds with minimal volume in up to 8 dimensions.

Contribution

It introduces explicit constructions of torsion-free subgroups in Coxeter groups via Weyl group actions, enabling the creation of small-volume hyperbolic manifolds.

Findings

Constructed torsion-free subgroups with small, explicitly determined index

Produced hyperbolic manifolds of minimal volume in dimensions up to 8

Established new connections between Weyl groups and geometric manifold construction

Abstract

By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction of hyperbolic manifolds of very small volume in up to 8 dimensions.