Embedding closed totally geodesic surfaces in Bianchi orbifolds

Bianchi and Hilbert-Blumenthal quaternionic orbifolds

In a series of papers, published in Mathematische Annalen, Bianchi and Blumenthal introduced the notions of Bianchi orbifolds and Hilbert-Blumnethal surfaces as generalizations of modular curves associated to quadratic fields. In this paper, in the same spirit, and following a similar line of reasoning, we introduce the concept of Bianchi and Hilbert-Blumenthal quaternionic orbifolds as generalizations of the Lipschitz and Hurwitz quaternionic modular orbifolds defined recently by Díaz, Vlacci and the first author. In particular, we describe the cusp cross-sections of the Hilbert-Blumenthal qu…

The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of techniques from analytic number theory, we address the following problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "s…

Areas of totally geodesic surfaces of hyperbolic 3-orbifolds

The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is a fundamental invariant of the topology of M via Mostow-Prasad Rigidity. Motivated by this, the second author and Reid defined a two-dimensional analogue of the geodesic length spectrum given by the multiset of isometry types of totally geodesic, immersed, finite-area surfaces of M called the geometric genus spectrum. They showed that if $M$ is arithmetic and contains a totally geodesic surface, then the geometric genus spectrum of M determines its commensurability class. In this paper we define a coarser inv…

On the convergence of arithmetic orbifolds

We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric ("Benjamini--Schramm") sense for hyperbolic three--space and a product of hyperbolic planes. We also deal with arbitrary sequences of maximal arithmetic three--dimensional hyperbolic lattices defined over a quadratic or cubic field. A motivating application is the study of Betti numbers of Bianchi groups.

超几何晶格中的子空间稳定子

Subspace stabilisers in hyperbolic lattices

本文研究超几何晶格中子空间的算术性判据，通过fc子空间的性质揭示轨道的算术特征。

$d = 2$ Bianchi群的极大富克斯子群

Maximal Fuchsian subgroups of the $d=2$ Bianchi group

研究$d = 2$的Bianchi群$\Gamma$的极大非初等富克斯子群共轭类，通过不定四元数代数整序给出明确描述并得余体积，还应用此计算了特定极限，涉及$\Gamma\backslash\mathbb{H}^3$中相关曲面数量。