Thin hyperbolic reflection groups

2021/12/29 by Bogachev, Nikolay, Kolpakov, Alexander
#20F55 #22E40 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2112.14642

Abstract

We study a family of Zariski dense finitely generated discrete subgroups of Isom(ℍ^d), d \geqslant 2, 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 non-reflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in Isom(ℍ^d), for any d \geqslant 2. 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.

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