arXiv research

Geodesics on modular surface yield arithmetic 3-manifolds.

The study of systoles in arithmetic hyperbolic manifolds.

Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.

3-manifolds study Hasse norm principle, akin to number fields.

We prove that if a closed hyperbolic 3-manifold M contains infinitely many totally geodesic surfaces, then M is arithmetic.

Proof of genus formula for 3-manifolds using arithmetic topology.

We will show that, for any noncompact arithmetic hyperbolic $m$-manifold with $m> 3$, and any compact arithmetic hyperbolic $m$-manifold with $m> 4$ that is not a $7$-dimensional arithmetic hyperbolic manifold defined by octonions, its fundamental group is not LERF. The main ingredient in the proof is a study on abelia…

Paper finds new 3D shapes that can be inside a 4D space.

We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently construct…

In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2,C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomol…

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce…

New lattices in higher rank contain a fixed 3-manifold group with increasing systole.

We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in …

Minhyong Kim introduced arithmetic Chern-Simons invariants for totally imaginary number fields as arithmetic analogues of the Chern-Simons invariants for 3-manifolds. In this paper, we extend Kim's definition for any number field, by using the modified étale cohomology groups and fundamental groups which take real plac…

In this paper we state and prove the analogous of the principal ideal theorem of algebraic number theory for the case of 3-manifolds from the point of view of arithmetic topology.

Formulates Hilbert reciprocity law on 3-manifolds.

In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum in which volume and 1-systole are controlled, and analyze th…

These are mostly expository notes based on the course of lectures on arithmetic invariants of hyperbolic manifolds given at the workshop associated with the final "Volume Conference," held at Columbia University, June 2009. Some new results are included.

We prove that, for any two finite volume hyperbolic $3$-manifolds, the amalgamation of their fundamental groups along any nontrivial geometrically finite subgroup is not LERF. This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic $3$-manifold groups along abelian subgroups. A consequ…

We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.

Arithmetic Dijkgraaf-Witten theory constructs analogues in Chern-Simons TQFT.

Let $M$ be a geometrically finite acylindrical hyperbolic 3-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$. These results were obtained earlier by McMullen, Moh…

In this paper we study the systole growth of arithmetic locally symmetric spaces up congruence covers and show that this growth is at least logarithmic in volume. This generalizes previous work of Buser and Sarnak as well as Katz, Schaps and Vishne where the case of compact hyperbolic 2- and 3-manifolds was considered.

New topological criterion extends arithmetic invariants in hyperbolic 3-manifolds.

We prove that a closed arithmetic hyperbolic 3-manifold with positive first betti number has virtually infinite first betti number.

We prove that there are only finitely many closed hyperbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements. An application to arithmetic manifolds is also given.

Löbell polyhedra have small systoles and are quasi-arithmetic.

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 tech…

Let M be an arithmetic hyperbolic 3-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of M, where each basis element is represented by a surface of `low' genus, and give evidence for this. We explain the relationship between this conjecture and the study of torsion homolo…

In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the commensurability class. In addition, we show that any finite volume hyperbolic 3-manifold …

This paper describes a general algorithm for finding the commensurator of a non-arithmetic cusped hyperbolic manifold, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this…

Mazur, Kapranov, Reznikov, and others developed ``Arithmetic Topology,'' a theory describing some surprising analogies between 3-dimensional topology and number theory, which can be summarized by saying that knots are like prime numbers. We extend their work by proving several formulas concerning branched coverings of …

The paper explores anomalous subvarieties in hyperbolic 3-manifolds and their geometric implications.

A Riemmanian foliated dynamical system of 3-dimension $(\mathrm{RFDS}^{3})$ is a closed Riemannian 3-manifold with additional structures: foliation, dynamical system. In the context of arithmetic topology, it is a geometric/analytic analogue of an arithmetic scheme with a conjectural dynamical system suggested by C. De…

The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in …

This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant …

In this note, we show that there exist cusped hyperbolic $3$-manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic $4$-manifolds, and…

Improved bounds linking entropy and volume in hyperbolic 3-manifolds.

We give estimates on the number $AL_H(x)$ of arithmetic lattices $Γ$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $…

We identify and study a class of hyperbolic 3-manifolds (which we call Macfarlane manifolds) whose quaternion algebras admit a geometric interpretation analogous to Hamilton's classical model for Euclidean rotations. We characterize these manifolds arithmetically, and show that infinitely many commensurability classes …

We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fix…

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 …

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds wit…

For a given cusped 3-manifold $M$ admitting an ideal triangulation, we describe a method to rigorously prove that either $M$ or a filling of $M$ admits a complete hyperbolic structure via verified computer calculations. Central to our method are an implementation of interval arithmetic and Krawczyk's Test. These techni…

Let $Γ$ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $Γ$ is arithmetic. This answers a question of Reid for hyperbolic $n$-manifolds and, independently, McMullen for hyperbolic $…

These are the extended notes of a talk I gave at the Geometric Topology Seminar of the Max Planck Institute for Mathematics in Bonn on January 30th, 2012. My goal was to familiarize the topologists with the basics of arithmetic hyperbolic 3-manifolds and sketch some interesting results in the theory of 3-manifolds (suc…