Arithmetic Fuchsian group


Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group. They, and the hyperbolic surface associated to their action on the hyperbolic plane often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.

Definition and examples

Quaternion algebras

A quaternion algebra over a field is a four-dimensional central simple -algebra. A quaternion algebra has a basis where and.
A quaternion algebra is said to be split over if it is isomorphic as an -algebra to the algebra of matrices.
If is an embedding of into a field we shall denote by the algebra obtained by extending scalars from to where we view as a subfield of via.

Arithmetic Fuchsian groups

A subgroup of is said to be derived from a quaternion algebra if it can be obtained through the following construction. Let be a totally real number field and a quaternion algebra over satisfying the following conditions. First there is a unique embedding such that is split over ; we denote by an isomorphism of -algebras. We also ask that for all other embeddings the algebra is not split. Next we need an order in. Let be the group of elements in of reduced norm 1 and let be its image in via. Then the image of is a subgroup of and we can consider the Fuchsian group which is its image in.
The main fact about these groups is that they are discrete subgroups and they have finite covolume for the Haar measure on Moreover, the construction above yields a cocompact subgroup if and only if the algebra is not split over. The discreteness is a rather immediate consequence of the fact that is only split at one real embedding. The finiteness of covolume is harder to prove.
An arithmetic Fuchsian group is any subgroup of which is commensurable to a group derived from a quaternion algebra. It follows immediately from this definition that arithmetic Fuchsian groups are discrete and of finite covolume.

Examples

The simplest example of an arithmetic Fuchsian group is the modular which is obtained by the construction above with and By taking Eichler orders in we obtain subgroups for of finite index in which can be explicitly written as follows:
Of course the arithmeticity of such subgroups follows from the fact that they are finite-index in the arithmetic group ; they belong to a more general class of finite-index subgroups, congruence subgroups.
Any order in a quaternion algebra over which is not split over but splits over yields a cocompact arithmetic Fuchsian group. There is a plentiful supply of such algebras.
More generally, all orders in quaternion algebras which are not yield cocompact subgroups. A further example of particular interest is obtained by taking to be the Hurwitz quaternions.

Maximal subgroups

A natural question is to identify those among arithmetic Fuchsian groups which are not strictly contained in a larger discrete subgroup. These are called maximal Kleinian groups and it is possible to give a complete classification in a given arithmetic commensurability class. Note that a theorem of Margulis implies that a lattice in is arithmetic if and only if it is commensurable to infinitely many maximal Kleinian groups.

Congruence subgroups

A principal congruence subgroup of is a subgroup of the form :
for some These are finite-index normal subgroups and the quotient is isomorphic to the finite group A congruence subgroup of is by definition a subgroup which contains a principal congruence subgroup.
Notably, not all finite-index subgroups of are congruence subgroups. A nice way to see this is to observe that has subgroups which surject onto the alternating group for arbitrary and since for large the group is not a subgroup of for any these subgroups cannot be congruence subgroups. In fact one can also see that there are many more non-congruence than congruence subgroups in.
The notion of a congruence subgroup generalizes to cocompact arithmetic Fuchsian groups and the results above also hold in this general setting.

Construction via quadratic forms

There is an isomorphism between and the connected component of the orthogonal group given by the action of the former by conjugation on the space of matrices of trace zero, on which the determinant induces the structure of a real quadratic space of signature. Arithmetic Fuchsian groups can be constructed directly in the latter group by taking the integral points in the orthogonal group associated to quadratic forms defined over number fields.
In this correspondence the modular group is associated up to commensurability to the group

Arithmetic Kleinian groups

The construction above can be adapted to obtain subgroups in : instead of asking for to be totally real and to be split at exactly one real embedding one asks for to have exactly one complex embedding up to complex conjugacy, at which is automatically split, and that is not split at any embedding. The subgroups of commensurable to those obtained by this construction are called arithmetic Kleinian groups. As in the Fuchsian case arithmetic Kleinian groups are discrete subgroups of finite covolume.

Trace fields of arithmetic Fuchsian groups

The invariant trace field of a Fuchsian group is the field generated by the traces of the squares of its elements. In the case of an arithmetic surface whose fundamental group is commensurable with a Fuchsian group derived from a quaternion algebra over a number field the invariant trace field equals.
One can in fact characterise arithmetic manifolds through the traces of the elements of their fundamental group, a result known as Takeuchi's criterion. A Fuchsian group is an arithmetic group if and only if the following three conditions are realised:
The Lie group is the group of positive isometries of the hyperbolic plane. Thus, if is a discrete subgroup of then acts properly discontinuously on. If moreover is torsion-free then the action is free and the quotient space is a surface with a hyperbolic metric. If is an arithmetic Fuchsian group such a surface is called an arithmetic hyperbolic surface. Since arithmetic Fuchsian groups are of finite covolume, arithmetic hyperbolic surfaces always have finite Riemannian volume.

Volume formula and finiteness

It is possible to give a formula for the volume of distinguished arithmetic surfaces from the arithmetic data with which it was constructed. Let be a maximal order in the quaternion algebra of discriminant over the field, let be its degree, its discriminant and its Dedekind zeta function. Let be the arithmetic group obtained from by the procedure above and the orbifold. Its volume is computed by the formula
the product is taken over prime ideals of dividing and we recall the is the norm function on ideals, i.e. is the cardinality of the finite ring ). The reader can check that if the output of this formula recovers the well-known result that the hyperbolic volume of the modular surface equals.
Coupled with the description of maximal subgroups and finiteness results for number fields this formula allows to prove the following statement:
Note that in dimensions four and more Wang's finiteness theorem asserts that this statement remains true by replacing "arithmetic" by "finite volume". An asymptotic equivalent for the number if arithmetic manifolds of a certain volume was given by Belolipetsky—Gelander—Lubotzky—Mozes.

Minimal volume

The hyperbolic orbifold of minimal volume can be obtained as the surface associated to a particular order, the Hurwitz quaternion order, and it is compact of volume.

Closed geodesics and injectivity radii

A closed geodesic on a Riemannian manifold is a closed curve that is also geodesic. One can give an effective description of the set of such curves in an arithmetic surface or three—manifold: they correspond to certain units in certain quadratic extensions of the base field. For example, the length of primitive closed geodesics in the modular surface corresponds to the absolute value of units of norm one in real quadratic fields. This description was used by Sarnak to establish a conjecture of Gauss on the mean order of class groups of real quadratic fields.
Arithmetic surfaces can be used to construct families of surfaces of genus for any which satisfy the systolic inequality

Spectra of arithmetic hyperbolic surfaces

Laplace eigenvalues and eigenfunctions

If is an hyperbolic surface then there is a distinguished operator on smooth functions on. In the case where is compact it extends to an unbounded, essentially self-adjoint operator on the Hilbert space of square-integrable functions on. The spectral theorem in Riemannian geometry states that there exists an orthonormal basis of eigenfunctions for. The associated eigenvalues are unbounded and their asymptotic behaviour is ruled by Weyl's law.
In the case where is arithmetic these eigenfunctions are a special type of automorphic forms for called Maass forms. The eigenvalues of are of interest for number theorists, as well as the distribution and nodal sets of the.
The case where is of finte volume is more complicated but a similar theory can be established via the notion of cusp form.

Selberg conjecture

The spectral gap of the surface is by definition the gap between the smallest eigenvalue and the second smallest eigenvalue ; thus its value equals and we shall denote it by In general it can be made arbitrarily small . The Selberg conjecture is the following statement providing a conjectural uniform lower bound in the arithmetic case:
Note that the statement is only valid for a subclass of arithmetic surfaces and can be seen to be false for general subgroups of finite index in lattices derived from quaternion algebras. Selberg's original statement was made only for congruence covers of the modular surface and it has been verified for some small groups. Selberg himself has proven the lower bound a result known as "Selberg's 1/16 theorem". The best known result in full generality is due to Luo—Rudnick—Sarnak.
The uniformity of the spectral gap has implications for the construction of expander graphs as Schreier graphs of

Relations with geometry

Selberg's trace formula shows that for an hyperbolic surface of finite volume it is equivalent to know the length spectrum and the spectrum of. However the precise relation is not explicit.
Another relation between spectrum and geometry is given by Cheeger's inequality, which in the case of a surface states roughly that a positive lower bound on the spectral gap of translates into a positive lower bound for the total length of a collection of smooth closed curves separating into two connected components.

Quantum ergodicity

The quantum ergodicity theorem of Shnirelman, Colin de Verdière and Zelditch states that on average, eigenfunctions equidistribute on. The unique quantum ergodicity conjecture of Rudnick and Sarnak asks whether the stronger statement that individual eigenfunctions equidistribure is true. Formally, the statement is as follows.
This conjecture has been proven by E. Lindenstrauss in the case where is compact and the are additionally eigenfunctions for the Hecke operators on. In the case of congruence covers of the modular some additional difficulties occur, which were dealt with by K. Soundararajan.

Isospectral surfaces

The fact that for arithmetic surfaces the arithmetic data determines the spectrum of the Laplace operator was pointed out by M. F. Vignéras and used by her to construct examples of isospectral compact hyperbolic surfaces. The precise statement is as follows:
Vignéras then constructed explicit instances for satisfying the conditions above and such that in addition is not conjugated by an element of to The resulting isospectral hyperbolic surfaces are then not isometric.