Counting and effective rigidity in algebra and geometry

TL;DR

This paper develops effective bounds for rigidity results in algebra and geometry, specifically relating to length spectra of hyperbolic manifolds and algebraic structures, providing explicit bounds based on volume.

Contribution

It introduces effective versions of rigidity theorems in hyperbolic geometry and algebra, with explicit bounds depending on volume and algebraic invariants.

Findings

Explicit bounds for length spectra differences in hyperbolic manifolds

Effective algebraic analogs involving quaternion algebras

Asymptotic behavior of counting functions 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.