Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using ordersinquaternion 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.
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
.
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 (this is equivalent to its being isomorphic to the Hamilton quaternions). 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
(since the reduced norm of a matrix algebra is just the determinant) 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 measureon 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.[1]
Anarithmetic 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 (this means that they are latticesin
).
The simplest example of an arithmetic Fuchsian group is the modular which is obtained by the construction above with
and
By taking Eichler ordersin
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.[2]
More generally, all orders in quaternion algebras (satisfying the above conditions) which are not yield cocompact subgroups. A further example of particular interest is obtained by taking
to be the Hurwitz quaternions.
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.
Aprincipal congruence subgroupof is a subgroup of the form :
for some These are finite-index normal subgroups and the quotient
is isomorphic to the finite group
Acongruence subgroupof
is by definition a subgroup which contains a principal congruence subgroup (these are the groups which are defined by taking the matrices in
which satisfy certain congruences modulo an integer, hence the name).
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
.[3]
The notion of a congruence subgroup generalizes to cocompact arithmetic Fuchsian groups and the results above also hold in this general setting.
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 (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 [4]
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.
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 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.[5] 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 discontinuouslyon
. If moreover
istorsion-free then the action is free and the quotient space
is a surface (a 2-manifold) with a hyperbolic metric (a Riemannian metric of constant sectional curvature −1). If
is an arithmetic Fuchsian group such a surface
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
of the volume form is finite).
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
ofdiscriminant
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[6]
the product is taken over prime idealsof 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 (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.[7]
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 .
Aclosed 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.[8]
Arithmetic surfaces can be used[9] to construct families of surfaces of genus for any
which satisfy the (optimal, up to a constant) systolic inequality
If is an hyperbolic surface then there is a distinguished operator
onsmooth functionson
. In the case where
is compact it extends to an unbounded, essentially self-adjoint operator on the Hilbert space
ofsquare-integrable functionson
. The spectral theorem in Riemannian geometry states that there exists an orthonormal basis
ofeigenfunctions 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.
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 (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[10] was made only for congruence covers of the modular surface and it has been verified for some small groups.[11] 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.[12]
The uniformity of the spectral gap has implications for the construction of expander graphs as Schreier graphs of [13]
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 , with multiplicities) 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.
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[14] in the case where is compact and the
are additionally eigenfunctions for the Hecke operatorson
. In the case of congruence covers of the modular some additional difficulties occur, which were dealt with by K. Soundararajan.[15]
The fact that for arithmetic surfaces the arithmetic data determines the spectrum of the Laplace operator was pointed out by M. F. Vignéras[16] 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.