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.

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

“…Additionally, they give an upper bound on the volume of their manifolds as a function of m. Inspired by [4], in this paper we consider the following question. This question was previously studied by the authors in [8]. Let π(V, S) denote the maximum cardinality of a collection of pairwise non-commensurable arithmetic hyperbolic 3-orbifolds derived from quaternion algebras, each of which has volume less than V and geodesic length spectrum containing S. In [8], it was shown that, if π(V, S) → ∞ as V → ∞, then there are integers 1 ≤ r, s ≤ |S| and constants c 1 , c 2 > 0 such that…”

Locally Equivalent Correspondences

“…For any subfield F ⊂ K, we have a homomorphism Res K/F : Br(F) → Br(K) given by Res K/F ([B]) = [B ⊗ F K]. For a pair of number fields K, K ′ , a natural isomorphism between Br(K), Br(K ′ ) is an isomorphism Φ Br : Br(K) → Br(K ′ ) such that for any F ⊂ K ∩ K ′ and any L with KK ′ ⊂ L, the diagram In [19] (see also [16]), it was shown that these fibers determine the algebra in certain situations. As in [20] however, there are situations when these fibers fail to determine the algebra.…”

Systolic surfaces of arithmetic hyperbolic 3-manifolds

“…Every totally geodesic surface is the area minimizing representative in its homotopy class, and we denote the set of all areas contributed by totally geodesic Our next theorem revisits a construction of pairs of incommensurable arithmetic hyperbolic 3manifolds with the same set of commensurability classes of surfaces up to an arbitrary threshold that appeared in [18,Section 7]. In particular we construct an infinite family of incommensurable arithmetic hyperbolic 3-manifolds whose geometric genus spectra start off with the same N terms and in which both volume and systole length are controlled.…”