Coordinates for quasi-Fuchsian punctured torus space

We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely the classical Fuchsian space with Fenchel-Nielsen coordinates and the Maskit embedding. We also show how they relate to the pleating coordinates of Keen and Series.

Prescribing the behaviour of geodesics in negative curvature

Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon desc…

On the representation of integers by indefinite binary Hermitian forms

Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most s by f, as s tends to infinity.

Bruhat–Tits Trees and Modular Groups

In this chapter, we give background information and preliminary results on the main link between the geometry and the algebra used for our arithmetic applications: the (discrete-time) geodesic ow on quotients of Bruhat{Tits trees by arithmetic lattices.

Integral binary Hamiltonian forms and their waterworlds

We give a graphical theory of integral indefinite binary Hamiltonian forms $f$ analogous to the one by Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order $\mathcal O$ in a definite quaternion algebra over $\mathbb Q$, we define the waterworld of $f$, analogous to Conway's river and Bestvina-Savin's ocean, and use it to give a combinatorial description of the values of $f$ on $\mathcal O\times\mathcal O$. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), and the $\operatorname{SL}_2(\mathcal O)$-equivariant Ford-Voronoi cellulation of the real …

Equidistribution of Common Perpendicular Arcs

In this chapter, we prove the equidistribution of the initial and terminal vectors of common perpendiculars of convex subsets, at the universal covering space level, for Riemannian manifolds and for metric and simplicial trees.

General conservation law for a class of physics field theories

In this paper we form a general conservation law that unifies a class of physics field theories. For this we first introduce the notion of a general field as a formal sum differential forms on a Minkowski manifold. Thereafter, we employ the action principle to define the conservation law for such general fields. By construction, particular field notions of physics, such as electric field strength, stress, strain etc. become instances of the general field. Hence, the differential equations that constitute physics field theories become also instances of the general conservation law. Accordingly, the general field and the general conservation law together correspond to a large class of physics…

Équidistribution non archimédienne et actions de groupes sur les arbres = Non-Archimedean equidistribution and group actions on trees

We give equidistribution results of elements of function fields over finite fields, and of quadratic irrationals over these fields, in their completed local fields. We deduce these results from equidistribution theorems of common perpendiculars in quotients of trees by lattices in their automorphism groups, proved by using ergodic properties of the discrete geodesic flow. Nous donnons des résultats d'équidistribution d'éléments de corps de fonctions sur des corps finis, et d'irrationnels quadratiques sur ces corps, dans leurs corps locaux complétés. Nous déduisons ces résultats de théorèmes d'équidistribution de perpendiculaires communes dans des quotients d'arbres par des réseaux de leur g…

Conformal Dehn surgery and the shape of Maskit’s embedding

We study the geometric limits of sequences of loxodromic cyclic groups which arise from conformal Dehn surgery. The results are applied to obtain an asymptotic description of the shape of the main cusp of the Maskit embedding of the Teichmüller space of once-punctured tori.

Rigidité, comptage et équidistribution de chaînes de Cartan quaternioniques

We prove an analog of Cartan's theorem, saying that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. We give a counting and equidistribution result for the orbits of arithmetic chains in the quaternionic Heisenberg group.; Nous montrons un analogue d'un théorème de Cartan, disant que les transformations préservant les chaînes sur le bord d'un espace hyperbolique quaternionien est une transformation projective. Nous donnons un résultat de comptage et d'équidistribution pour une orbite de chaînes arithmétiques dans le groupe de Heisenberg quaternionique.

Equidistribution of Equidistant Level Sets to Gibbs Measures

Before stating this equidistribution result, we begin with a technical construction that will also be useful in the following chapter.

Appendix: Diophantine Approximation on Hyperbolic Surfaces

In this (independent) appendix, we study the Diophantine approximation properties for the particular case of the cusped hyperbolic surfaces, in the spirit of Sect. 2 (or [11]), and the many still open questions that arise for them. We refer to [9], [10]for fundamental results and further developments. We study in particular the distance to a cusp of closed geodesics on a hyperbolic surface.

Counting and equidistribution in quaternionic Heisenberg groups

AbstractWe develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

Negatively Curved Geometry

Let X be a geodesically complete proper CAT(–1) space, let x0 ∈ X be an arbitrary basepoint, and let Γ be a nonelementary discrete group of isometries of X.

Rate of Mixing for the Geodesic Flow

The main part of the chapter then consists in proving analogous bounds for the discrete-time and continuous-time geodesic ow for quotient spaces of simplicial and metric trees respectively.

A classification of $\protect \mathbb{R}$-Fuchsian subgroups of Picard modular groups

A classification of C-Fuchsian subgroups of Picard modular groups

Equidistribution and Counting of Quadratic Irrational Points in Non-Archimedean Local Fields

We use these results to deduce equidistribution and counting results of quadratic irrational elements in non-Archimedean local fields.

Symbolic Dynamics of Geodesic Flows on Trees

In this chapter, we give a coding of the discrete-time geodesic ow on the nonwandering sets of quotients of locally finite simplicial trees X without terminal vertices by nonelementary discrete subgroups of Aut(X) by a subshift of finite type on a countable alphabet.

On the nonarchimedean quadratic Lagrange spectra

We study Diophantine approximation in completions of functions fields over finite fields, and in particular in fields of formal Laurent series over finite fields. We introduce a Lagrange spectrum for the approximation by orbits of quadratic irrationals under the modular group. We give nonarchimedean analogs of various well known results in the real case: the closedness and boundedness of the Lagrange spectrum, the existence of a Hall ray, as well as computations of various Hurwitz constants. We use geometric methods of group actions on Bruhat-Tits trees. peerReviewed

Equidistribution and Counting of Cross-ratios

The following properties of relative heights are easy to check using the definitions, the invariance properties of the cross-ratio, and Equation (17.1).

Potentials, Critical Exponents,and Gibbs Cocycles

Let X be a geodesically complete proper CAT(–1) space, let x0 ∈ X be an arbitrary basepoint, and let Γ be a nonelementary discrete group of isometries of X.

Counting and equidistribution in Heisenberg groups

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal and isotropic for a given Hermitian form over $K$, and a counting and equidistribution result for …

Explicit Measure Computations for Simplicial Trees and Graphs of Groups

In this chapter, we compute skinning measures and Bowen{Margulis measures for some highly symmetric simplicial trees X endowed with a nonelementary discrete subgroup Г of Aut(X).

Counting common perpendicular arcs in negative curvature

Let $D^-$ and $D^+$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ from $D^-$ to $D^+$, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^-$ and $D^+$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with finite volume, we give an error term for the asymptotic. As an application, we give an asymptotic form…

Rate of Mixing for Equilibrium States in Negative Curvature and Trees

In this survey based on the recent book by the three authors, we recall the Patterson-Sullivan construction of equilibrium states for the geodesic flow on negatively curved orbifolds or tree quotients, and discuss their mixing properties, emphasizing the rate of mixing for (not necessarily compact) tree quotients via coding by countable (not necessarily finite) topological shifts. We give a new construction of numerous nonuniform tree lattices such that the (discrete time) geodesic flow on the tree quotient is exponentially mixing with respect to the maximal entropy measure: we construct examples whose tree quotients have an arbitrary space of ends or an arbitrary (at most exponential) grow…

Équidistribution non archimédienne et actions de groupes sur les arbres

Resume Nous donnons des resultats d'equidistribution d'elements de corps de fonctions sur des corps finis, et d'irrationnels quadratiques sur ces corps, dans leurs corps locaux completes. Nous deduisons ces resultats de theoremes d'equidistribution de perpendiculaires communes dans des quotients d'arbres par des reseaux de leur groupe d'automorphismes, demontres a l'aide de proprietes ergodiques du flot geodesique discret.

Geometric limits of cyclic groups and the shape of Schottky space

We give a local estimate of the shape of Schottky space at certain boundary points. This is done by studying the geometric limits of sequences of cyclic groups and applying the results to the generators of Schottky groups. We also obtain information on discrete, non-faithful representations close to the cusps.

Equidistribution and Counting of Integral Representations by Quadratic Norm Forms

In the final chapter of this book, we give another arithmetic equidistribution and counting result of rational elements in non-Archimedean local fields of positive characteristic, again using our geometric equidistribution and counting results of common perpendiculars in trees summarised in Section 15.4.

Equidistribution and Counting of Rational Points in Completed Function Fields

Let K be a (global) function field over Fq of genus g, let v be a (normalised discrete) valuation of K, let Kv be the associated completion of K, and let Rv be the affine function ring associated with v.

Patterson–Sullivan and Bowen–Margulis Measures with Potential on CAT(–1) Spaces

In this chapter, we discuss geometrically and dynamically relevant measures on the boundary at infinity of X and on the space of geodesic lines gX.

A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature

This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in \(\mathbb{R}\), \(\mathbb{C}\) and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition…

Skinning Measures with Potential on CAT(–1) Spaces

In this chapter, we introduce skinning measures as weighted pushforwards of the Patterson{Sullivan densities associated with a potential to the unit normal bundles of convex subsets of a CAT(–1) space.

Equidistribution and Counting of Common Perpendiculars in Quotient Spaces

In this chapter, we use the results of Chapter 11 to prove equidistribution and counting results in Riemannian manifolds (or good orbifolds) and in metric and simplicial graphs (of groups).

On the statistics of pairs of logarithms of integers

We study the statistics of pairs of logarithms of positive integers at various scalings, either with trivial weights or with weights given by the Euler function, proving the existence of pair correlation functions. We prove that at the linear scaling, which is not the usual scaling by the inverse of the average gap, the pair correlations exhibit a level repulsion similar to radial distribution functions of fluids. We prove total loss of mass phenomena at superlinear scalings, and constant nonzero asymptotic behavior at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp nei…

Random Walks on Weighted Graphs of Groups

Let X be a locally finite simplicial tree without terminal vertices, and let X = ∣X∣1 be its geometric realisation. Let Γ be a nonelementary discrete subgroup of Aut(X).

Fields with Discrete Valuations

In the present chapter, before embarking on our arithmetic applications, we recall basic facts on local fields for the convenience of the geometer reader.

Sur les rayons de Hall en approximation diophantienne

Resume Nous montrons que l'existence d'un rayon de Hall dans le spectre de Lagrange des constantes d'approximation d'un nombre reel par des nombres rationnels se generalise a de nombreux problemes d'approximation diophantienne, comme consequence de la possibilite de prescrire arbitrairement une hauteur de penetration asymptotique suffisamment grande d'une geodesique dans un voisinage d'une pointe d'une variete riemannienne de volume fini a courbure strictement negative. Pour citer cet article : J. Parkkonen, F. Paulin, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

On the arithmetic and geometry of binary Hamiltonian forms

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.