Dotti Edoardo, Drewitz Simon T, Kellerhals Ruth
Department of Mathematics, University of Fribourg, 1700 Fribourg, Switzerland.
Discrete Comput Geom. 2023;69(3):873-895. doi: 10.1007/s00454-022-00455-z. Epub 2022 Nov 25.
For three distinct infinite families , , and of non-arithmetic 1-cusped hyperbolic Coxeter 3-orbifolds, we prove incommensurability for a pair of elements and belonging to the same sequence and for most pairs belonging two different ones. We investigate this problem first by means of the Vinberg space and the Vinberg form, a quadratic space associated to each of the corresponding fundamental Coxeter prism groups, which allows us to deduce some partial results. The complete proof is based on the analytic behavior of another commensurability invariant. It is given by the cusp density, and we prove and exploit its strict monotonicity.
对于三个不同的非算术单尖点双曲Coxeter 3 - 轨形的无限族(\mathcal{O}_1)、(\mathcal{O}_2)和(\mathcal{O}_3)，我们证明了属于同一序列的一对元素(\gamma)和(\delta)以及大多数属于两个不同序列的元素对的不可公度性。我们首先借助Vinberg空间和Vinberg形式来研究这个问题，Vinberg形式是与每个相应的基本Coxeter棱柱群相关联的二次空间，这使我们能够推导出一些部分结果。完整的证明基于另一个可公度性不变量的解析行为。它由尖点密度给出，并且我们证明并利用了它的严格单调性。
