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62-608: This is related to the congruence subgroup Γ ( 1 ) {\displaystyle \Gamma (1)} in the following way: Here the j -invariant is normalized such that j = 0 {\displaystyle j=0} has complex multiplication by Z [ ζ 3 ] {\displaystyle \mathbb {Z} [\zeta _{3}]} , and j = 1728 {\displaystyle j=1728} has complex multiplication by Z [ i ] {\displaystyle \mathbb {Z} [i]} . The j -line can be seen as giving

124-405: A Cauchy sequence to comply with in the congruence topology than in the profinite topology). The congruence kernel C ( Γ ) {\displaystyle C(\Gamma )} is the kernel of this morphism, and the congruence subgroup problem stated above amounts to whether C ( Γ ) {\displaystyle C(\Gamma )} is trivial. The weakening of

186-496: A basis of open neighbourhoods of the identity, the collection of all normal subgroups of finite index in G . The resulting topology is called the profinite topology on G . A group is residually finite if, and only if, its profinite topology is Hausdorff . A group whose cyclic subgroups are closed in the profinite topology is said to be Π C {\displaystyle \Pi _{C}\,} . Groups each of whose finitely generated subgroups are closed in

248-427: A congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible 2 × 2 integer matrices of determinant 1 in which the off-diagonal entries are even . More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups ; that is, those for which we have

310-455: A sesquilinear form over the Hamilton quaternions), plus the exceptional group F 4 − 20 {\displaystyle F_{4}^{-20}} (see List of simple Lie groups ). The current status of the congruence subgroup problem is as follows: In many situations where the congruence subgroup problem is expected to have a positive solution it has been proven that this

372-465: A coordinatization of the classical modular curve of level 1, X 0 ( 1 ) {\displaystyle X_{0}(1)} , which is isomorphic to the complex projective line P / C 1 {\displaystyle \mathbb {P} _{/\mathbb {C} }^{1}} . This mathematics -related article is a stub . You can help Misplaced Pages by expanding it . Congruence subgroup In mathematics ,

434-482: A faithful representation ⁠ ρ {\displaystyle \rho } ⁠ , also defined over ⁠ Q {\displaystyle \mathbb {Q} } ⁠ , from G {\displaystyle \mathbf {G} } into ⁠ G L d {\displaystyle \mathrm {GL} _{d}} ⁠ ; then an arithmetic group in G ( Q ) {\displaystyle \mathbf {G} (\mathbb {Q} )}

496-406: A family of expander graphs . There is also a representation-theoretical interpretation: if Γ {\displaystyle \Gamma } is a lattice in a Lie group ⁠ G {\displaystyle G} ⁠ then property (τ) is equivalent to the non-trivial unitary representations of ⁠ G {\displaystyle G} ⁠ occurring in

558-711: A fundamental example of automorphic forms . Other automorphic forms associated to these congruence subgroups are the holomorphic modular forms, which can be interpreted as cohomology classes on the associated Riemann surfaces via the Eichler-Shimura isomorphism . The normalizer Γ 0 ( p ) + {\displaystyle \Gamma _{0}(p)^{+}} of Γ 0 ( p ) {\displaystyle \Gamma _{0}(p)} in S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )} has been investigated; one result from

620-472: A given arithmetic group Γ {\displaystyle \Gamma } always has property (τ) of Lubotzky–Zimmer. This can be taken to mean that the Cheeger constant of the family of their Schreier coset graphs (with respect to a fixed generating set for ⁠ Γ {\displaystyle \Gamma } ⁠ ) is uniformly bounded away from zero, in other words they are

682-520: A notion of 'integral structure' and can define reduction maps modulo an integer. The existence of congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is residually finite . An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem , which asks whether all subgroups of finite index are essentially congruence subgroups. Congruence subgroups of 2 × 2 matrices are fundamental objects in

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744-657: A positive solution: its origin is in the work of Hyman Bass , Jean-Pierre Serre and John Milnor , and Jens Mennicke who proved that, in contrast to the case of S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )} , when n ⩾ 3 {\displaystyle n\geqslant 3} all finite-index subgroups in S L n ( Z ) {\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )} are congruence subgroups. The solution by Bass–Milnor–Serre involved an aspect of algebraic number theory linked to K-theory . On

806-483: A principal congruence subgroup (a notion that does not depend on a representation). They are subgroups of finite index that correspond to the subgroups of the finite groups ⁠ π n ( Γ ) {\displaystyle \pi _{n}(\Gamma )} ⁠ , and the level is defined. The principal congruence subgroups of S L d ( Z ) {\displaystyle \mathrm {SL} _{d}(\mathbb {Z} )} are

868-437: A principal congruence subgroup of Γ {\displaystyle \Gamma } to be the kernel of π n {\displaystyle \pi _{n}} (which may a priori depend on the representation ⁠ ρ {\displaystyle \rho } ⁠ ), and a congruence subgroup of Γ {\displaystyle \Gamma } to be any subgroup that contains

930-433: A subgroup of ⁠ G ( A f ) {\displaystyle \mathbf {G} (\mathbb {A} _{f})} ⁠ , in particular the congruence completion Γ ¯ {\displaystyle {\overline {\Gamma }}} is its closure in that group. These remarks are also valid for ⁠ S {\displaystyle S} ⁠ -arithmetic subgroups, replacing

992-484: Is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, or 71. When Ogg later heard about the monster group , he noticed that these were precisely the prime factors of the size of ⁠ M {\displaystyle M} ⁠ , he wrote up a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact – this was a starting point for the theory of monstrous moonshine , which explains deep connections between modular function theory and

1054-404: Is a Q {\displaystyle \mathbb {Q} } -group and S = { p 1 , … , p r } {\displaystyle S=\{p_{1},\ldots ,p_{r}\}} is a finite set of primes, an S {\displaystyle S} -arithmetic subgroup of G ( Q ) {\displaystyle \mathbf {G} (\mathbb {Q} )}

1116-441: Is a congruence subgroup if and only if its closure H ¯ ⊂ G ( A f ) {\displaystyle {\overline {H}}\subset \mathbf {G} (\mathbb {A} _{f})} is a compact-open subgroup (compactness is automatic) and ⁠ H = Γ ∩ H ¯ {\displaystyle H=\Gamma \cap {\overline {H}}} ⁠ . In general

1178-540: Is a topology on Γ {\displaystyle \Gamma } for which a base of neighbourhoods of the trivial subgroup is the set of subgroups of finite index (the profinite topology ); and there is another topology defined in the same way using only congruence subgroups. The profinite topology gives rise to a completion ⁠ Γ ^ {\displaystyle {\widehat {\Gamma }}} ⁠ of ⁠ Γ {\displaystyle \Gamma } ⁠ , while

1240-566: Is an integer not divisible by any prime in ⁠ S {\displaystyle S} ⁠ , then all primes p i {\displaystyle p_{i}} are invertible modulo n {\displaystyle n} and it follows that there is a morphism ⁠ π n : Γ S → G L d ( Z / n Z ) {\displaystyle \pi _{n}:\Gamma _{S}\to \mathrm {GL} _{d}(\mathbb {Z} /n\mathbb {Z} )} ⁠ . Thus it

1302-794: Is an integer there is a homomorphism π n : S L 2 ( Z ) → S L 2 ( Z / n Z ) {\displaystyle \pi _{n}:\mathrm {SL} _{2}(\mathbb {Z} )\to \mathrm {SL} _{2}(\mathbb {Z} /n\mathbb {Z} )} induced by the reduction modulo n {\displaystyle n} morphism ⁠ Z → Z / n Z {\displaystyle \mathbb {Z} \to \mathbb {Z} /n\mathbb {Z} } ⁠ . The principal congruence subgroup of level n {\displaystyle n} in Γ = S L 2 ( Z ) {\displaystyle \Gamma =\mathrm {SL} _{2}(\mathbb {Z} )}

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1364-425: Is any group Γ ⊂ G ( Q ) {\displaystyle \Gamma \subset \mathbf {G} (\mathbb {Q} )} that is of finite index in the stabiliser of a finite-index sub-lattice in ⁠ Z d {\displaystyle \mathbb {Z} ^{d}} ⁠ . Let Γ {\displaystyle \Gamma } be an arithmetic group: for simplicity it

1426-844: Is at least 2: The cases of inner and outer forms of type A n {\displaystyle A_{n}} are still open. The algebraic groups in the case of inner forms of type A n {\displaystyle A_{n}} are those associated to the unit groups in central simple division algebras; for example the congruence subgroup property is not known for lattices in S L 3 ( R ) {\displaystyle \mathrm {SL} _{3}(\mathbb {R} )} or S L 2 ( R ) × S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )\times \mathrm {SL} _{2}(\mathbb {R} )} with compact quotient. The ring of adeles A {\displaystyle \mathbb {A} }

1488-627: Is better to suppose that ⁠ Γ ⊂ G L n ( Z ) {\displaystyle \Gamma \subset \mathrm {GL} _{n}(\mathbb {Z} )} ⁠ . As in the case of S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )} there are reduction morphisms ⁠ π n : Γ → G L d ( Z / n Z ) {\displaystyle \pi _{n}:\Gamma \to \mathrm {GL} _{d}(\mathbb {Z} /n\mathbb {Z} )} ⁠ . We can define

1550-974: Is defined as an arithmetic subgroup but using Z [ 1 / p 1 , … , 1 / p r ] ) {\displaystyle \mathbb {Z} [1/p_{1},\ldots ,1/p_{r}])} instead of ⁠ Z {\displaystyle \mathbb {Z} } ⁠ . The fundamental example is ⁠ SL d ⁡ ( Z [ 1 / p 1 , … , 1 / p r ] ) {\displaystyle \operatorname {SL} _{d}(\mathbb {Z} [1/p_{1},\ldots ,1/p_{r}])} ⁠ . Let Γ S {\displaystyle \Gamma _{S}} be an S {\displaystyle S} -arithmetic group in an algebraic group ⁠ G ⊂ GL d {\displaystyle \mathbf {G} \subset \operatorname {GL} _{d}} ⁠ . If n {\displaystyle n}

1612-425: Is indeed the case. Here is a list of algebraic groups such that the congruence subgroup property is known to hold for the associated arithmetic lattices, in case the rank of the associated Lie group (or more generally the sum of the rank of the real and ⁠ p {\displaystyle p} ⁠ -adic factors in the case of ⁠ S {\displaystyle S} ⁠ -arithmetic groups)

1674-439: Is injective. This implies the following result: The group Γ ( 2 ) {\displaystyle \Gamma (2)} contains − Id {\displaystyle -\operatorname {Id} } and is not torsion-free. On the other hand, its image in PSL 2 ⁡ ( Z ) {\displaystyle \operatorname {PSL} _{2}(\mathbb {Z} )}

1736-427: Is natural to ask whether they account for all finite-index subgroups in ⁠ Γ {\displaystyle \Gamma } ⁠ . The answer is a resounding "no". This fact was already known to Felix Klein and there are many ways to exhibit many non-congruence finite-index subgroups. For example: One can ask the same question for any arithmetic group as for the modular group: This problem can have

1798-523: Is of index 3 and is explicitly described by: These subgroups satisfy the following inclusions: ⁠ Γ ( n ) ⊂ Γ 1 ( n ) ⊂ Γ 0 ( n ) {\displaystyle \Gamma (n)\subset \Gamma _{1}(n)\subset \Gamma _{0}(n)} ⁠ , as well as ⁠ Γ ( 2 ) ⊂ Λ {\displaystyle \Gamma (2)\subset \Lambda } ⁠ . The congruence subgroups of

1860-449: Is possible to define congruence subgroups in ⁠ Γ S {\displaystyle \Gamma _{S}} ⁠ , whose level is always coprime to all primes in ⁠ S {\displaystyle S} ⁠ . Congruence subgroups in Γ = S L 2 ( Z ) {\displaystyle \Gamma =\mathrm {SL} _{2}(\mathbb {Z} )} are finite-index subgroups: it

1922-531: Is prime then Γ / Γ 0 ( p ) {\displaystyle \Gamma /\Gamma _{0}(p)} is in natural bijection with the projective line over the finite field ⁠ F p {\displaystyle \mathbb {F} _{p}} ⁠ , and explicit representatives for the (left or right) cosets of Γ 0 ( p ) {\displaystyle \Gamma _{0}(p)} in Γ {\displaystyle \Gamma } are

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1984-465: Is residually finite. In particular, all profinite groups are residually finite. Examples of non-residually finite groups can be constructed using the fact that all finitely generated residually finite groups are Hopfian groups . For example the Baumslag–Solitar group B (2,3) is not Hopfian, and therefore not residually finite. Every group G may be made into a topological group by taking as

2046-549: Is the restricted product of all completions of ⁠ Q {\displaystyle \mathbb {Q} } ⁠ , i.e. where the product is over the set P {\displaystyle {\mathcal {P}}} of all primes, Q p {\displaystyle \mathbb {Q} _{p}} is the field of p -adic numbers and an element ( x , ( x p ) p ∈ P ) {\displaystyle (x,(x_{p})_{p\in {\mathcal {P}}})} belongs to

2108-470: Is the ring of integers in a number field , for example ⁠ O = Z [ 2 ] {\displaystyle O=\mathbb {Z} [{\sqrt {2}}]} ⁠ . Then if p {\displaystyle {\mathfrak {p}}} is a prime ideal dividing a rational prime p {\displaystyle p} the subgroups Γ ( p ) {\displaystyle \Gamma ({\mathfrak {p}})} that

2170-581: Is the kernel of ⁠ π n {\displaystyle \pi _{n}} ⁠ , and it is usually denoted ⁠ Γ ( n ) {\displaystyle \Gamma (n)} ⁠ . Explicitly it is described as follows: This definition immediately implies that Γ ( n ) {\displaystyle \Gamma (n)} is a normal subgroup of finite index in ⁠ Γ {\displaystyle \Gamma } ⁠ . The strong approximation theorem (in this case an easy consequence of

2232-1089: Is the kernel of the reduction map mod p {\displaystyle {\mathfrak {p}}} is a congruence subgroup since it contains the principal congruence subgroup defined by reduction modulo ⁠ p {\displaystyle p} ⁠ . Yet another arithmetic group is the Siegel modular groups ⁠ S p 2 g ( Z ) {\displaystyle \mathrm {Sp} _{2g}(\mathbb {Z} )} ⁠ , defined by: Note that if g = 1 {\displaystyle g=1} then ⁠ S p 2 ( Z ) = S L 2 ( Z ) {\displaystyle \mathrm {Sp} _{2}(\mathbb {Z} )=\mathrm {SL} _{2}(\mathbb {Z} )} ⁠ . The theta subgroup Γ ϑ ( n ) {\displaystyle \Gamma _{\vartheta }^{(n)}} of S p 2 g ( Z ) {\displaystyle \mathrm {Sp} _{2g}(\mathbb {Z} )}

2294-666: Is the preimage of the subgroup of unipotent matrices: Their indices are given by the formula: The theta subgroup Λ {\displaystyle \Lambda } is the congruence subgroup of Γ {\displaystyle \Gamma } defined as the preimage of the cyclic group of order two generated by ( 0 − 1 1 0 ) ∈ S L 2 ( Z / 2 Z ) {\displaystyle \left({\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}\right)\in \mathrm {SL} _{2}(\mathbb {Z} /2\mathbb {Z} )} . It

2356-544: Is the set of all ( A B C D ) ∈ S p 2 g ( Z ) {\displaystyle \left({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}}\right)\in \mathrm {Sp} _{2g}(\mathbb {Z} )} such that both A B ⊤ {\displaystyle AB^{\top }} and C D ⊤ {\displaystyle CD^{\top }} have even diagonal entries. The family of congruence subgroups in

2418-502: Is torsion-free, and the quotient of the hyperbolic plane by this subgroup is a sphere with three cusps. A subgroup H {\displaystyle H} in Γ = S L 2 ( Z ) {\displaystyle \Gamma =\mathrm {SL} _{2}(\mathbb {Z} )} is called a congruence subgroup if there exists n ⩾ 1 {\displaystyle n\geqslant 1} such that H {\displaystyle H} contains

2480-510: The Chinese remainder theorem ) implies that π n {\displaystyle \pi _{n}} is surjective, so that the quotient Γ / Γ ( n ) {\displaystyle \Gamma /\Gamma (n)} is isomorphic to ⁠ S L 2 ( Z / n Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} /n\mathbb {Z} )} ⁠ . Computing

2542-508: The Hecke congruence subgroup of level ⁠ n {\displaystyle n} ⁠ , is defined as the preimage by π n {\displaystyle \pi _{n}} of the group of upper triangular matrices. That is, The index is given by the formula: where the product is taken over all prime numbers dividing ⁠ n {\displaystyle n} ⁠ . If p {\displaystyle p}

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2604-523: The adelic algebraic group G ( A ) {\displaystyle \mathbf {G} (\mathbb {A} )} is well-defined. It can be endowed with a canonical topology, which in the case where G {\displaystyle \mathbf {G} } is a linear algebraic group is the topology as a subset of ⁠ A m {\displaystyle \mathbb {A} ^{m}} ⁠ . The finite adèles A f {\displaystyle \mathbb {A} _{f}} are

2666-710: The mathematical field of group theory , a group G is residually finite or finitely approximable if for every element g that is not the identity in G there is a homomorphism h from G to a finite group , such that There are a number of equivalent definitions: Examples of groups that are residually finite are finite groups , free groups , finitely generated nilpotent groups , polycyclic-by-finite groups , finitely generated linear groups , and fundamental groups of compact 3-manifolds . Subgroups of residually finite groups are residually finite, and direct products of residually finite groups are residually finite. Any inverse limit of residually finite groups

2728-514: The orthogonal groups ⁠ S O ( d , 1 ) , d ⩾ 2 {\displaystyle \mathrm {SO} (d,1),d\geqslant 2} ⁠ , the unitary groups S U ( d , 1 ) , d ⩾ 2 {\displaystyle \mathrm {SU} (d,1),d\geqslant 2} and the groups S p ( d , 1 ) , d ⩾ 2 {\displaystyle \mathrm {Sp} (d,1),d\geqslant 2} (the isometry groups of

2790-415: The real rank of G {\displaystyle G} is at least 2; for example, lattices in S L 3 ( R ) {\displaystyle \mathrm {SL} _{3}(\mathbb {R} )} should always have the property. Serre's conjecture states that a lattice in a Lie group of rank one should not have the congruence subgroup property. There are three families of such groups:

2852-456: The "congruence" topology gives rise to another completion ⁠ Γ ¯ {\displaystyle {\overline {\Gamma }}} ⁠ . Both are profinite groups and there is a natural surjective morphism Γ ^ → Γ ¯ {\displaystyle {\widehat {\Gamma }}\to {\overline {\Gamma }}} (intuitively, there are fewer conditions for

2914-520: The 1970s, due to Jean-Pierre Serre , Andrew Ogg and John G. Thompson is that the corresponding modular curve (the Riemann surface resulting from taking the quotient of the hyperbolic plane by ⁠ Γ 0 ( p ) + {\displaystyle \Gamma _{0}(p)^{+}} ⁠ ) has genus zero (i.e., the modular curve is a Riemann sphere) if and only if ⁠ p {\displaystyle p} ⁠

2976-478: The classical theory of modular forms ; the modern theory of automorphic forms makes a similar use of congruence subgroups in more general arithmetic groups. The simplest interesting setting in which congruence subgroups can be studied is that of the modular group ⁠ S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )} ⁠ . If n ⩾ 1 {\displaystyle n\geqslant 1}

3038-421: The conclusion then leads to the following problem. When the problem has a positive solution one says that Γ {\displaystyle \Gamma } has the congruence subgroup property . A conjecture generally attributed to Serre states that an irreducible arithmetic lattice in a semisimple Lie group G {\displaystyle G} has the congruence subgroup property if and only if

3100-466: The discrete subgroup ⁠ G ( Q ) ⊂ G ( A ) {\displaystyle \mathbf {G} (\mathbb {Q} )\subset \mathbf {G} (\mathbb {A} )} ⁠ . This is especially convenient in the theory of automorphic forms: for example all modern treatments of the Arthur–Selberg trace formula are done in this adélic setting. Residually finite In

3162-718: The following matrices: The subgroups Γ 0 ( n ) {\displaystyle \Gamma _{0}(n)} are never torsion-free as they always contain the matrix ⁠ − I {\displaystyle -I} ⁠ . There are infinitely many n {\displaystyle n} such that the image of Γ 0 ( n ) {\displaystyle \Gamma _{0}(n)} in P S L 2 ( Z ) {\displaystyle \mathrm {PSL} _{2}(\mathbb {Z} )} also contains torsion elements. The subgroup Γ 1 ( n ) {\displaystyle \Gamma _{1}(n)}

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3224-410: The group Γ ∩ H ¯ {\displaystyle \Gamma \cap {\overline {H}}} is equal to the congruence closure of H {\displaystyle H} in ⁠ Γ {\displaystyle \Gamma } ⁠ , and the congruence topology on Γ {\displaystyle \Gamma } is the induced topology as

3286-493: The modular group and the associated Riemann surfaces are distinguished by some particularly nice geometric and topological properties. Here is a sample: There is also a collection of distinguished operators called Hecke operators on smooth functions on congruence covers, which commute with each other and with the Laplace–Beltrami operator and are diagonalisable in each eigenspace of the latter. Their common eigenfunctions are

3348-464: The monster group. The notion of an arithmetic group is a vast generalisation based upon the fundamental example of ⁠ S L d ( Z ) {\displaystyle \mathrm {SL} _{d}(\mathbb {Z} )} ⁠ . In general, to give a definition one needs a semisimple algebraic group G {\displaystyle \mathbf {G} } defined over Q {\displaystyle \mathbb {Q} } and

3410-454: The order of this finite group yields the following formula for the index: where the product is taken over all prime numbers dividing ⁠ n {\displaystyle n} ⁠ . If n ⩾ 3 {\displaystyle n\geqslant 3} then the restriction of π n {\displaystyle \pi _{n}} to any finite subgroup of Γ {\displaystyle \Gamma }

3472-470: The other hand, the work of Serre on S L 2 {\displaystyle \mathrm {SL} _{2}} over number fields shows that in some cases the answer to the naïve question is "no" while a slight relaxation of the problem has a positive answer. This new problem is better stated in terms of certain compact topological groups associated to an arithmetic group ⁠ Γ {\displaystyle \Gamma } ⁠ . There

3534-508: The principal congruence subgroup ⁠ Γ ( n ) {\displaystyle \Gamma (n)} ⁠ . The level l {\displaystyle l} of H {\displaystyle H} is then the smallest such ⁠ n {\displaystyle n} ⁠ . From this definition it follows that: The subgroup ⁠ Γ 0 ( n ) {\displaystyle \Gamma _{0}(n)} ⁠ , sometimes called

3596-458: The restricted product if and only if for almost all primes ⁠ p {\displaystyle p} ⁠ , x p {\displaystyle x_{p}} belongs to the subring Z p {\displaystyle \mathbb {Z} _{p}} of p -adic integers . Given any algebraic group G {\displaystyle \mathbf {G} } over Q {\displaystyle \mathbb {Q} }

3658-397: The restricted product of all non-archimedean completions (all p -adic fields). If Γ ⊂ G ( Q ) {\displaystyle \Gamma \subset \mathbf {G} (\mathbb {Q} )} is an arithmetic group then its congruence subgroups are characterised by the following property: H ⊂ Γ {\displaystyle H\subset \Gamma }

3720-740: The ring of finite adèles with the restricted product over all primes not in ⁠ S {\displaystyle S} ⁠ . More generally one can define what it means for a subgroup Γ ⊂ G ( Q ) {\displaystyle \Gamma \subset \mathbf {G} (\mathbb {Q} )} to be a congruence subgroup without explicit reference to a fixed arithmetic subgroup, by asking that it be equal to its congruence closure ⁠ Γ ¯ ∩ G ( Q ) {\displaystyle {\overline {\Gamma }}\cap \mathbf {G} (\mathbb {Q} )} ⁠ . Thus it becomes possible to study all congruence subgroups at once by looking at

3782-531: The spaces L 2 ( G / Γ ) {\displaystyle L^{2}(G/\Gamma )} being bounded away from the trivial representation (in the Fell topology on the unitary dual of ⁠ G {\displaystyle G} ⁠ ). Property (τ) is a weakening of Kazhdan's property (T) which implies that the family of all finite-index subgroups has property (τ). If G {\displaystyle \mathbf {G} }

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3844-526: The subgroups Γ ( n ) {\displaystyle \Gamma (n)} given by: the congruence subgroups then correspond to the subgroups of S L d ( Z / n Z ) {\displaystyle \mathrm {SL} _{d}(\mathbb {Z} /n\mathbb {Z} )} . Another example of arithmetic group is given by the groups S L 2 ( O ) {\displaystyle \mathrm {SL} _{2}(O)} where O {\displaystyle O}

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