Semi-Arithmetic Fuchsian Groups and Modular Embeddings

In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.

Length spectra of periodic orbits are investigated for some chaotic dynamical systems whose quantum energy spectra show unexpected statistical properties and for which the notion of arithmetical chaos has been introduced recently. These systems are defined as the unconstrained motions of particles on two-dimensional surfaces of constant negative curvature whose fundamental groups are given by number theoretical statements (arithmetic Fuchsian groups). It is shown that the mean multiplicity of lengths l of periodic orbits grows asymptotically like c.e12//l,I to infinity ., Moreover, the constant c (depending on the arithmetic group) is determined.

According to a theorem of Belyi, a smooth projective algebraic curve is defined over a number field if and only if there exists a non-constant element of its function field ramified only over 0, 1 and ∞. The existence of such a Belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the Poincaré upper half plane by a subgroup of finite index in a Fuchsian triangle group. On the other hand, Fuchsian triangle groups arise in many contexts, such as in the theory of hypergeometric functions and certain triangular billiard problems, which would appear at first sight to have no relation to the Galois problems that motivated the above discovery of Belyi. In this note we review several results related to Belyi's theorem and we develop certain aspects giving examples. For preliminary accounts, see the preprint [Wo1], the conference proceedings article [BauItz] and the “Comptes Rendus” note [CoWo2].

We construct a Teichmüller curve uniformized by a Fuchsian triangle group commensurable to .m;n; 1/ for every m; n Ä 1.In most cases, for example when m ¤ n and m or n is odd, the uniformizing group is equal to the triangle group .m;n; 1/.Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases.The construction essentially relies on properties of hypergeometric differential operators.For small m, we find billiard tables that generate these Teichmüller curves.We interpret some of the so-called Lyapunov exponents of the Kontsevich-Zorich cocycle as normalized degrees of a natural line bundle on a Teichmüller curve.We determine the Lyapunov exponents for the Teichmüller curves we construct.

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 and apply our criteria to study the effect of abstract field automorphisms of $\mathbb{C}$ on modular embeddings. Finally we derive that the absolute Galois group of $\mathbb{Q}$ operates on the dessins d'enfants defined by principal congruence subgroups of (arithmetic or non-arithmetic) triangle groups by permuting the defining ideals in the tautological way.

Let G be a Fuchsian group containing two torsion free subgroups defining isomorphic Riemann surfaces. Then these surface subgroups K and αKα−1 are conjugate in PSL(2,R), but in general the conjugating element α cannot be taken in G or a finite index Fuchsian extension of G. We will show that in the case of a normal inclusion in a triangle group G these α can be chosen in some triangle group extending G. It turns out that the method leading to this result allows also to answer the question of how many different regular dessins of the same type can exist on a given quasiplatonic Riemann surface.

We show that the Halphen transform of a Lamé equation can be written as the symmetric square of the Lamé equation followed by an Euler transform. We use this to compute a list of Lamé equations with arithmetic Fuchsian monodromy group. It contains all those Lamé equations where the quaternion algebra A associated to the arithmetic Fuchsian group is defined over ℚ.

This chapter describes methods for computing explicit classifying spaces and free resolutions for a range of discrete groups. These are illustrated using computer examples involving: aspherical groups, graphs of groups, special linear groups, triangle groups, generalized triangle groups, Coxeter groups, Artin groups, and arithmetic groups.

The work of Nikulin and Agol, Belolipetsky, Storm, and Whyte shows that only finitely many number fields may serve as the field of definition of an arithmetic hyperbolic reflection group. An important problem posed by Nikulin is to enumerate these fields and their degrees. In this paper we prove that in dimension 2 the degree of these fields is at most 7. More generally we prove that the degree of the field of definition of the quaternion algebra associated to an arithmetic Fuchsian group of genus 0 is at most 7, confirming a conjecture of Long, Maclachlan and Reid. We also obtain upper bounds for the discriminants of these fields of definition, allowing for an enumeration which should be useful for the classification of arithmetic hyperbolic reflection groups.

Let $$p$$ and $$\ell $$ be two distinct prime numbers and let $$\Gamma $$ be a group. We study the asymptotic behaviour of the mod- $$\ell $$ Betti numbers in $$p$$ -adic analytic towers of finite index subgroups. If $$\Theta $$ is a finite $$\ell $$ -group of automorphisms of $$\Gamma $$ , our main theorem allows to lift lower bounds for the mod- $$\ell $$ cohomology growth in the fixed point group $$\Gamma ^\Theta $$ to lower bounds for the growth in $$\Gamma $$ . We give applications to $$S$$ -arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.

Rational Period Functions and Parabolic Cohomology

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 on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of “Bolza twins” (using magma). Like the Hurwitz triplets, these correspond to the factoring of certain rational primes in the ring of integers of the invariant trace field of the surface. We exploit random sampling combined with the Reidemeister–Schreier algorithm as implemented in magma to generate these surfaces.

For a certain diagonal flow on SL(d,Z)﹨SL(d,R) where d≥2, we show that any bounded subset (with measure zero boundary) of the horosphere or a translated horosphere equidistributes, under a suitable normalization, on a target shrinking into the cusp. This type of equidistribution is shrinking target horospherical equidistribution (STHE), and we show STHE for several types of shrinking targets. Our STHE results extend known results for d=2 and L﹨PSL(2,R) where L is any cofinite Fuchsian group with at least one cusp.The two key tools needed to prove our STHE results for the horosphere are a renormalization technique and Marklof's result on the equidistribution of the Farey sequence on distinguished sections. For our STHE results for translated horospheres, we introduce translated Farey sequences, develop some of their geometric and dynamical properties, generalize Marklof's result by proving the equidistribution of translated Farey sequences for the same distinguished sections, and use this equidistribution of translated Farey sequences along with the renormalization technique to prove our STHE results for translated horospheres.

The first section of this chapter gives a short account of uniformisation theory for Riemann surfaces, in which the main result is the extended Riemann mapping theorem. We describe the classification of cocompact Fuchsian groups, and the local and global properties of quotients of the hyperbolic plane by such groups. We discuss the inclusion relations between these groups, classified by Singerman, giving particular attention to triangle groups. These groups turn out to be very important, not only for a better understanding of maps and hypermaps, but also because certain automorphic functions related to them act as a source for all Belyĭ functions. These ideas are illustrated with Klein’s quartic curve, a classic example of a Riemann surface of genus 3. In the second section we use triangle groups to prove Grothendieck’s important observation, that even a purely topological definition of a dessin as a graph embedded in a compact oriented surface leads to a unique conformal structure on this surface, so that the dessin corresponds to a Belyĭ function on the resulting Riemann surface. This shows that our previous definitions of dessins (via Belyĭ functions, embedded graphs or permutations) are in fact equivalent. Grothendieck attributed this result to Malgoire and Voisin, and in the meantime there have been many proofs, such as that given by Voevodsky and Shabat, but here we use ideas of Singerman from pre-dessins times, involving maps and triangle groups. The chapter includes an appendix explaining group presentations.

In a 2016 paper by Alan Reid, Martin Bridson and the author, it was shown using the theory of profinite groups that if $\Gamma$ is a finitely-generated Fuchsian group and $\Sigma$ is a lattice in a connected Lie group, such that $\Gamma$ and $\Sigma$ have exactly the same finite quotients, then $\Gamma$ is isomorphic to $\Sigma$. As a consequence, two triangle groups $\Delta(r,s,t)$ and $\Delta(u,v,w)$ have the same finite quotients if and only if $(u,v,w)$ is a permutation of $(r,s,t)$. A direct proof of this property of triangle groups was given in the final section of that paper, with the purpose of exhibiting explicit finite quotients that can distinguish one triangle group from another. Unfortunately, part of the latter direct proof was flawed. In this paper two new direct proofs are given, one being a corrected version using the same approach as before (involving direct products of small quotients), and the other being a shorter one that uses the same preliminary observations as in the earlier version but then takes a different direction (involving further use of the `Macbeath trick').