Thin hyperbolic reflection groups

Abstract We study a family of Zariski dense finitely generated discrete subgroups of , , defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an application, we obtain a general description of all thin hyperbolic reflection groups. In particular, we show that the Vinberg algorithm applied to a nonreflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in , for any . Moreover, every such group is a subgroup of a group produced by the Vinberg algorithm applied to a Lorentzian lattice independently on the latter being reflective. As a consequence, all thin hyperbolic reflection groups are enumerable.