arXiv research

Proves quaternionic analog of Cartan's theorem and counts arithmetic chains.

Study counts and equidistributes rational points in quaternionic Heisenberg groups.

New bounds on diameters and generators for specific lattices and graphs.

We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry.

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…

Using the rings of Lipschitz and Hurwitz integers $\mathbb{H}(\mathbb{Z})$ and $\mathbb{H}ur(\mathbb{Z})$ in the quaternion division algebra $\mathbb{H}$, we define several Kleinian discrete subgroups of $PSL(2,\mathbb{H})$

This thesis investigates cusp cross-sections of arithmetic real, complex, and quaternionic hyperbolic $n$--orbifolds. We give a smooth classification of these submanifolds and analyze their induced geometry. One of the primary tools is a new subgroup separability result for general arithmetic lattices.

We classify all torsion-free derived arithmetic Fuchsian groups of genus two by commensurability class. In particular, we show that there exist no such groups arising from quaternion algebras over number fields of degree greater than 5. We also prove some results on the existence and form of maximal orders for a class …

Study on OI surfaces with unique geometric properties.

We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such gr…

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $Γ$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of $Γ$ is studied in some detail.

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). Th…

Extends shape analysis to framed space curves using quaternionic arithmetic.

Computes presentations for specific arithmetic hyperbolic lattices.

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…

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 …

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 analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times …

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient co…

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…

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 …

Study polynomial trace identities in $SL(2,\IC)$ using quaternion algebras.

The purpose of this paper is to give presentations for projective $S$-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field $\mathbb{Q}$. To our knowledge, this provides the first explicit presentations of an $S$-arithmetic lattice in a semisimple Lie group with $S$ large. In particular, we…

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b$. A special case describes all Shimura subvarieties of type $\mathrm{A}_1$ Shimura varieties. We produce, for any $n\geq 1…

The paper proves properties of Brauer equivalent number fields and applies them to geometric spaces.

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group a…

Building on results of Arthur and Mok, we extend to (finite volume) complex and quaternionic hyperbolic manifolds the results of arXiv:1004.1085. For the spherical spectrum our results are optimal. Finally, as an application we prove a Lefschetz property for the restriction map between arithmetic quotients of complex b…

Paper extends algebraic geometry results to hyperbolic link complements.

Let $M$ be a compact 3-manifold and $Γ=π_1(M)$. Work of Thurston and Culler--Shalen established the $\mathrm{SL}_2(\mathbb{C})$ character variety $X(Γ)$ as fundamental tool in the study of the geometry and topology of $M$. This is particularly the case when $M$ is the exterior of a hyperbolic knot $K$ in $S^3$. The mai…

Let $k$ be a number field and $B$ be a central simple algebra over $k$ of dimension $p^2$ where $p$ is prime. In the case that $p=2$ we assume that $B$ is not totally definite. In this paper we study sets of pairwise nonisomorphic maximal orders of $B$ with the property that a $\mathcal{O}_k$-order of rank $p$ embeds i…

We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in $SL(1,D)$, where $D$ is a quaternion division algebras defined over a number field $E$ contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbol…

Study of quaternion Azumaya algebras in hyperbolic once-punctured torus bundles.

Computes canonical heights for arithmetic log surfaces using Hurwitz zeta function.

Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.

We study the covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$ for $n\geq 2$ and identify uniform and non-uniform irreducible lattices of minimal covolume. More precisely, let $μ$ be the Euler-Poincaré measure on $PSL_2(\mathbb R)^n$ and $χ=μ/2^n$. We show that the Hilbert modular group $PSL_2(\mathfrak o_{k_{49…

In this paper, we give the definitions and characterizations of quaternionic Salkowski, quaternionic anti-Salkowski and quaternionic similar curves in the Euclidean spaces E^3 and E^4. We obtain relationships between these curves and some special quaternionic curves such as quaternionic slant helices and quaternionic B…

New adaptive filters reduce energy consumption in electronic devices.

We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. W…

The study finds conditions for quaternionic structures on symmetric spaces.

We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic p…

We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) A map between quaternion…

Defines quaternionic k-vector fields on quaternionic Kähler manifolds.

The paper proves quaternion projective space is unstable.

The paper studies quaternionic structures on GKM graphs and their relation to torus actions on quaternionic projective spaces.

Penrose's two-spinor notation for $4$-dimensional Lorentzian manifolds can be extended to two-component notation for quaternionic manifolds, which is a very useful tool for calculation. We construct a family of quaternionic complexes over unimodular quaternionic manifolds by elementary calculation. On complex quaternio…

We introduce the notion of CR quaternionic map and we prove that any such real-analytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold $M$, of dimension $4k-1$, of a quaternionic manifold…

The paper extends Gray's result to quaternion-Kähler manifolds.