Counting and effective rigidity in algebra and geometry

Abstract

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.

v.2, 39 pages. To appear in Invent. Math

References in corpus (7)

• Weakly commensurable arithmetic groups, lengths of closed geodesics and isospectral locally symmetric spaces
• Division Algebras and Non-Commensurable Isospectral Manifolds
• Geometric spectra and commensurability
• Grunwald- Wang Theorem, an Effective Version
• Spectrally similar incommensurable 3-manifolds
• Families of mutually isospectral Riemannian orbifolds
• The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces

Cited by in corpus (4)

• Division algebras with the same maximal subfields
• Strong Limit Multiplicity for arithmetic hyperbolic surfaces and $3$-manifolds
• Spectrally similar incommensurable 3-manifolds
• Asymptotics of extensions of simple $\mathbb{Q}$-algebras