In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example
SL2(\Z).
One of the origins of the mathematical theory of arithmetic groups is algebraic number theory. The classical reduction theory of quadratic and Hermitian forms by Charles Hermite, Hermann Minkowski and others can be seen as computing fundamental domains for the action of certain arithmetic groups on the relevant symmetric spaces.[1] [2] The topic was related to Minkowski's geometry of numbers and the early development of the study of arithmetic invariant of number fields such as the discriminant. Arithmetic groups can be thought of as a vast generalisation of the unit groups of number fields to a noncommutative setting.
The same groups also appeared in analytic number theory as the study of classical modular forms and their generalisations developed. Of course the two topics were related, as can be seen for example in Langlands' computation of the volume of certain fundamental domains using analytic methods. This classical theory culminated with the work of Siegel, who showed the finiteness of the volume of a fundamental domain in many cases.
For the modern theory to begin foundational work was needed, and was provided by the work of Armand Borel, André Weil, Jacques Tits and others on algebraic groups.[3] [4] Shortly afterwards the finiteness of covolume was proven in full generality by Borel and Harish-Chandra.[5] Meanwhile, there was progress on the general theory of lattices in Lie groups by Atle Selberg, Grigori Margulis, David Kazhdan, M. S. Raghunathan and others. The state of the art after this period was essentially fixed in Raghunathan's treatise, published in 1972.[6]
In the seventies Margulis revolutionised the topic by proving that in "most" cases the arithmetic constructions account for all lattices in a given Lie group.[7] Some limited results in this direction had been obtained earlier by Selberg, but Margulis' methods (the use of ergodic-theoretical tools for actions on homogeneous spaces) were completely new in this context and were to be extremely influential on later developments, effectively renewing the old subject of geometry of numbers and allowing Margulis himself to prove the Oppenheim conjecture; stronger results (Ratner's theorems) were later obtained by Marina Ratner.
In another direction the classical topic of modular forms has blossomed into the modern theory of automorphic forms. The driving force behind this effort is mainly the Langlands program initiated by Robert Langlands. One of the main tool used there is the trace formula originating in Selberg's work[8] and developed in the most general setting by James Arthur.[9]
Finally arithmetic groups are often used to construct interesting examples of locally symmetric Riemannian manifolds. A particularly active research topic has been arithmetic hyperbolic 3-manifolds, which as William Thurston wrote,[10] "...often seem to have special beauty."
If
G
GLn(\Q)
n
G(\Q)
\Gamma=GLn(\Z)\capG(\Q).
\Q
G\toGLn(\Q).
Thus a better notion is to take for definition of an arithmetic subgroup of
G(\Q)
Λ
\Gamma/(\Gamma\capΛ)
Λ/(\Gamma\capΛ)
\Gamma
GLn
G
A natural generalisation of the construction above is as follows: let
F
O
G
F
\rho:G\toGLn
F
\rho-1(GLn(O))\subsetG(F)
On the other hand, the class of groups thus obtained is not larger than the class of arithmetic groups as defined above. Indeed, if we consider the algebraic group
G'
\Q
F
\Q
\Q
\rho':G'\toGLdn
\rho
d=[F:\Q]
(\rho')-1(GLnd(\Z))
The classical example of an arithmetic group is
SLn(\Z)
PSLn(\Z)
GLn(\Z)
PGLn(\Z)
n=2
PSL2(\Z)
SL2(\Z)
Sp2g(\Z)
Other well-known and studied examples include the Bianchi groups
SL2(O-m),
m>0
O-m
\Q(\sqrt{-m}),
SL2(Om)
Another classical example is given by the integral elements in the orthogonal group of a quadratic form defined over a number field, for example
SO(n,1)(\Z)
When
G
G
G
\Q
G(\R)\toG
G(\Q)
G
G=G(\R)
G
GLn
G\capGLn(\Z)
G
SLn(\Z)
SLn(\R)
A lattice in a Lie group is usually defined as a discrete subgroup with finite covolume. The terminology introduced above is coherent with this, as a theorem due to Borel and Harish-Chandra states that an arithmetic subgroup in a semisimple Lie group is of finite covolume (the discreteness is obvious).
The theorem is more precise: it says that the arithmetic lattice is cocompact if and only if the "form" of
G
\Q
G
n
\Q
\Qn\setminus\{0\}
The spectacular result that Margulis obtained is a partial converse to the Borel—Harish-Chandra theorem: for certain Lie groups any lattice is arithmetic. This result is true for all irreducible lattice in semisimple Lie groups of real rank larger than two.[11] [12] For example, all lattices in
SLn(\R)
n\ge3
Irreducibility only plays a role when
G
G=G1 x G2
Gi
SL2(\Z[\sqrt2])
SL2(\R) x SL2(\R)
SL2(\Z) x SL2(\Z)
The Margulis arithmeticity (and superrigidity) theorem holds for certain rank 1 Lie groups, namely
Sp(n,1)
n\geqslant1
-20 | |
F | |
4 |
SO(n,1)
n\geqslant2
SU(n,1)
n=1,2,3
SU(n,1)
n\geqslant4
See main article: Arithmetic Fuchsian group.
See main article: Arithmetic hyperbolic 3-manifold.
F
A
F
lO
A
\sigma:F\to\R
A\sigma ⊗ F\R
M2(\R)
lO1
(A\sigma ⊗ F\R)1
SL2(\R),
A
\Q.
SL2(\R)
Arithmetic Kleinian groups are constructed similarly except that
F
A
SL2(\Complex).
For every semisimple Lie group
G
G
G=SL2(\R),SL2(\Complex)
G
See main article: Congruence subgroup.
A congruence subgroup is (roughly) a subgroup of an arithmetic group defined by taking all matrices satisfying certain equations modulo an integer, for example the group of 2 by 2 integer matrices with diagonal (respectively off-diagonal) coefficients congruent to 1 (respectively 0) modulo a positive integer. These are always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way. The conjecture (usually attributed to Jean-Pierre Serre) is that this is true for (irreducible) arithmetic lattices in higher-rank groups and false in rank-one groups. It is still open in this generality but there are many results establishing it for specific lattices (in both its positive and negative cases).
Instead of taking integral points in the definition of an arithmetic lattice one can take points which are only integral away from a finite number of primes. This leads to the notion of an
S
S
SL2\left(\Z\left[\tfrac1p\right]\right)
SL2\left(\Z\left[\tfrac1p\right]\right)
SL2(\R) x SL2(\Qp).
The formal definition of an
S
S
GLn(\Z)
GLn\left(\Z\left[\tfrac1N\right]\right)
N
S
The Borel–Harish-Chandra theorem generalizes to
S
\Gamma
S
\Q
G
\Gamma
G=G(\R) x \prodp\inG(\Qp)
Arithmetic groups with Kazhdan's property (T) or the weaker property (
\tau
Congruence covers of arithmetic surfaces are known to give rise to surfaces with large injectivity radius. Likewise the Ramanujan graphs constructed by Lubotzky-Phillips-Sarnak have large girth. It is in fact known that the Ramanujan property itself implies that the local girths of the graph are almost always large.[18]
Arithmetic groups can be used to construct isospectral manifolds. This was first realised by Marie-France Vignéras[19] and numerous variations on her construction have appeared since. The isospectrality problem is in fact particularly amenable to study in the restricted setting of arithmetic manifolds.[20]
P2(\Complex)
PU(2,1)