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 formula_1. 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 formula_2 is a four-dimensional central simple formula_2-algebra. A quaternion algebra has a basis formula_4 where formula_5 and formula_6.
A quaternion algebra is said to be split over formula_2 if it is isomorphic as an formula_2-algebra to the algebra of matrices formula_9.
If formula_10 is an embedding of formula_2 into a field formula_12 we shall denote by formula_13 the algebra obtained by extending scalars from formula_2 to formula_12 where we view formula_2 as a subfield of formula_12 via formula_10.
Arithmetic Fuchsian groups.
A subgroup of formula_19 is said to be "derived from a quaternion algebra" if it can be obtained through the following construction. Let formula_2 be a totally real number field and formula_21 a quaternion algebra over formula_2 satisfying the following conditions. First there is a unique embedding formula_23 such that formula_24 is split over formula_25 ; we denote by formula_26 an isomorphism of formula_25-algebras. We also ask that for all other embeddings formula_28 the algebra formula_29 is not split (this is equivalent to its being isomorphic to the Hamilton quaternions). Next we need an order formula_30 in formula_21. Let formula_32 be the group of elements in formula_30 of reduced norm 1 and let formula_34 be its image in formula_35 via formula_36. Then the image of formula_34 is a subgroup of formula_38 (since the reduced norm of a matrix algebra is just the determinant) and we can consider the Fuchsian group which is its image in formula_19.
The main fact about these groups is that they are discrete subgroups and they have finite covolume for the Haar measure on formula_40 Moreover, the construction above yields a cocompact subgroup if and only if the algebra formula_21 is not split over formula_2. The discreteness is a rather immediate consequence of the fact that formula_21 is only split at one real embedding. The finiteness of covolume is harder to prove.
An "arithmetic Fuchsian group" is any subgroup of formula_19 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 (this means that they are lattices in formula_19).
Examples.
The simplest example of an arithmetic Fuchsian group is the modular formula_46 which is obtained by the construction above with formula_47 and formula_48 By taking Eichler orders in formula_21 we obtain subgroups formula_50 for formula_51 of finite index in formula_52 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 formula_54 ; they belong to a more general class of finite-index subgroups, congruence subgroups.
Any order in a quaternion algebra over formula_55 which is not split over formula_55 but splits over formula_25 yields a cocompact arithmetic Fuchsian group. There is a plentiful supply of such algebras.
More generally, all orders in quaternion algebras (satisfying the above conditions) which are not formula_58 yield cocompact subgroups. A further example of particular interest is obtained by taking formula_21 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 formula_60 is arithmetic if and only if it is commensurable to infinitely many maximal Kleinian groups.
Congruence subgroups.
A "principal congruence subgroup" of formula_61 is a subgroup of the form :
for some formula_63 These are finite-index normal subgroups and the quotient formula_64 is isomorphic to the finite group formula_65 A "congruence subgroup" of formula_34 is by definition a subgroup which contains a principal congruence subgroup (these are the groups which are defined by taking the matrices in formula_34 which satisfy certain congruences modulo an integer, hence the name).
Notably, not all finite-index subgroups of formula_68 are congruence subgroups. A nice way to see this is to observe that formula_68 has subgroups which surject onto the alternating group formula_70 for arbitrary formula_71 and since for large formula_72 the group formula_70 is not a subgroup of formula_74 for any formula_75 these subgroups cannot be congruence subgroups. In fact one can also see that there are many more non-congruence than congruence subgroups in formula_68.
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 formula_19 and the connected component of the orthogonal group formula_78 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 (2,1). 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 (and satisfying certain conditions).
In this correspondence the modular group is associated up to commensurability to the group formula_79
Arithmetic Kleinian groups.
The construction above can be adapted to obtain subgroups in formula_60: instead of asking for formula_2 to be totally real and formula_21 to be split at exactly one real embedding one asks for formula_2 to have exactly one complex embedding up to complex conjugacy, at which formula_21 is automatically split, and that formula_21 is not split at any embedding formula_86. The subgroups of formula_60 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 (or, through the monodromy image of the fundamental group, of a hyperbolic surface) 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 formula_2 the invariant trace field equals formula_2.
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:
Geometry of arithmetic hyperbolic surfaces.
The Lie group formula_19 is the group of positive isometries of the hyperbolic plane formula_97. Thus, if formula_98 is a discrete subgroup of formula_19 then formula_34 acts properly discontinuously on formula_97. If moreover formula_34 is torsion-free then the action is free and the quotient space formula_103 is a surface (a 2-manifold) with a hyperbolic metric (a Riemannian metric of constant sectional curvature −1). If formula_34 is an arithmetic Fuchsian group such a surface formula_105 is called an "arithmetic hyperbolic surface" (not to be confused with the arithmetic surfaces from arithmetic geometry; however when the context is clear the "hyperbolic" specifier may be omitted). Since arithmetic Fuchsian groups are of finite covolume, arithmetic hyperbolic surfaces always have finite Riemannian volume (i.e. the integral over formula_105 of the volume form is finite).
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 formula_30 be a maximal order in the quaternion algebra formula_21 of discriminant formula_109 over the field formula_2, let formula_111 be its degree, formula_112 its discriminant and formula_113 its Dedekind zeta function. Let formula_114 be the arithmetic group obtained from formula_30 by the procedure above and formula_105 the orbifold formula_117. Its volume is computed by the formula
the product is taken over prime ideals of formula_119 dividing formula_120 and we recall the formula_121 is the norm function on ideals, i.e. formula_122 is the cardinality of the finite ring formula_123). The reader can check that if formula_124 the output of this formula recovers the well-known result that the hyperbolic volume of the modular surface equals formula_125.
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 (a consequence of local rigidity) 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 formula_128.
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 (the description is lengthy and shall not be given in full here). 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 formula_129 for any formula_129 which satisfy the (optimal, up to a constant) systolic inequality
Spectra of arithmetic hyperbolic surfaces.
Laplace eigenvalues and eigenfunctions.
If formula_105 is an hyperbolic surface then there is a distinguished operator formula_133 on smooth functions on formula_105. In the case where formula_105 is compact it extends to an unbounded, essentially self-adjoint operator on the Hilbert space formula_136 of square-integrable functions on formula_105. The spectral theorem in Riemannian geometry states that there exists an orthonormal basis formula_138 of eigenfunctions for formula_133. The associated eigenvalues formula_140 are unbounded and their asymptotic behaviour is ruled by Weyl's law.
In the case where formula_141 is arithmetic these eigenfunctions are a special type of automorphic forms for formula_34 called Maass forms. The eigenvalues of formula_133 are of interest for number theorists, as well as the distribution and nodal sets of the formula_144.
The case where formula_105 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 formula_105 is by definition the gap between the smallest eigenvalue formula_147 and the second smallest eigenvalue formula_148; thus its value equals formula_149 and we shall denote it by formula_150 In general it can be made arbitrarily small (ref Randol) (however it has a positive lower bound for a surface with fixed volume). 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 formula_156 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 formula_157
Relations with geometry.
Selberg's trace formula shows that for an hyperbolic surface of finite volume it is equivalent to know the length spectrum (the collection of lengths of all closed geodesics on formula_105, with multiplicities) and the spectrum of formula_133. 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 formula_105 states roughly that a positive lower bound on the spectral gap of formula_105 translates into a positive lower bound for the total length of a collection of smooth closed curves separating formula_105 into two connected components.
Quantum ergodicity.
The quantum ergodicity theorem of Shnirelman, Colin de Verdière and Zelditch states that on average, eigenfunctions equidistribute on formula_105. 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 formula_105 is compact and the formula_165 are additionally eigenfunctions for the Hecke operators on formula_105. 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 formula_133 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 formula_180 satisfying the conditions above and such that in addition formula_181 is not conjugated by an element of formula_21 to formula_183. The resulting isospectral hyperbolic surfaces are then not isometric.