Counting and effective rigidity in algebra and geometry

• 成果类型: Article
• 署名作者: Linowitz, Benjamin; McReynolds, D. B.; Pollack, Paul; Thompson, Lola
• 署名单位: University System of Ohio; Oberlin College; Purdue University System; Purdue University; University System of Georgia; University of Georgia
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0796-y
• 发表日期: 2018
• 页码: 697-758
• 关键词: arithmetic hyperbolic 3-manifolds; quaternion algebras; commensurability classes; isospectral manifolds; division-algebras; subgroup growth; kleinian-groups; length spectra; number-fields; lie-groups

摘要

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.