Thin hyperbolic reflection groups

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-06-08 DOI:10.1112/blms.70108

Nikolay Bogachev, Alexander Kolpakov

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Abstract

We study a family of Zariski dense finitely generated discrete subgroups of $Isom (H^{d})$ , $d ⩾ 2$ , 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 nonreflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in $Isom (H^{d})$ , for any $d ⩾ 2$ . 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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薄双曲反射群

我们研究了Isom (H d) $\mathrm{Isom}(\mathbb {H}^d)$的一个Zariski密集有限生成离散子群族。D小于或等于2 $d \geqslant 2$，由以下属性定义：这个家族中的任何组在超平面中至少包含一个反射。作为应用，我们得到了所有薄双曲反射群的一般描述。特别地，我们证明了将Vinberg算法应用于非反射洛伦兹晶格会在Isom (H d) $\mathrm{Isom}(\mathbb {H}^d)$中产生无限序列的薄反射子群。对于任何d小于2 $d \geqslant 2$。而且，每一个这样的群都是由单独应用于洛伦兹格的Vinberg算法产生的群的子群。因此，所有薄双曲反射群都是可枚举的。

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