Weyl groups, lattices and geometric manifolds. (Q735042)

The paper presents an algebraic construction of torsion free subgroups (both normal and non-normal) of small index in Coxeter groups which satisfy an admissibility condition introduced by the authors. The method is based on a study of the action of the Weyl group of a simple Lie algebra on its root lattice. The construction is then applied to certain non-cocompact hyperbolic reflection groups in dimensions \(n=4\), \(6\) and \(8\) where it allows to produce hyperbolic \(n\)-manifolds of very small volume. In particular, in dimensions \(4\) and \(6\) it gives rise to manifolds whose volume is only twice the volume of the smallest non-compact hyperbolic manifolds in these dimensions.