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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an arithmetic group is a group obtained as the integer points of an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
s and other classical topics in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...
. They also give rise to very interesting examples of
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s and hence are objects of interest in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. Finally, these two topics join in the theory of
automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s which is fundamental in modern number theory.


History

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 Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermi ...
,
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
and others can be seen as computing
fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
s for the action of certain arithmetic groups on the relevant
symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, l ...
s. The topic was related to Minkowski's
geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental informatio ...
and the early development of the study of arithmetic invariant of number fields such as the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
. Arithmetic groups can be thought of as a vast generalisation of the
unit group In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this ...
s 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 Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
,
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
,
Jacques Tits Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life and ...
and others on algebraic groups. Shortly afterwards the finiteness of covolume was proven in full generality by Borel and Harish-Chandra. Meanwhile, there was progress on the general theory of lattices in Lie groups by
Atle Selberg Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded t ...
, Grigori Margulis,
David Kazhdan David Kazhdan ( he, דוד קשדן), born Dmitry Aleksandrovich Kazhdan (russian: Дми́трий Александро́вич Кажда́н), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 Ma ...
,
M. S. Raghunathan Madabusi Santanam Raghunathan FRS is an Indian mathematician. He is currently Head of the National Centre for Mathematics, Indian Institute of Technology, Mumbai. Formerly Professor of eminence at TIFR in Homi Bhabha Chair. Raghunathan receiv ...
and others. The state of the art after this period was essentially fixed in Raghunathan's treatise, published in 1972. 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. 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 In Diophantine approximation, the Oppenheim conjecture concerns representations of numbers by real quadratic forms in several variables. It was formulated in 1929 by Alexander Oppenheim and later the conjectured property was further strengthened b ...
; stronger results (
Ratner's theorems In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study ...
) were later obtained by
Marina Ratner Marina Evseevna Ratner (russian: Мари́на Евсе́евна Ра́тнер; October 30, 1938 – July 7, 2017) was a professor of mathematics at the University of California, Berkeley who worked in ergodic theory. Around 1990, she proved ...
. 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 In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
initiated by
Robert Langlands Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study o ...
. One of the main tool used there is the trace formula originating in Selberg's work and developed in the most general setting by
James Arthur James Arthur (born 2 March 1988) is an English singer and songwriter. He rose to fame after winning the ninth series of ''The X Factor'' in 2012. His debut single, a cover of Shontelle's " Impossible", was released by Syco Music after the fi ...
. 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 William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...
wrote, "...often seem to have special beauty."


Definition and construction


Arithmetic groups

If \mathrm G is an algebraic subgroup of \mathrm_n(\Q) for some n then we can define an arithmetic subgroup of \mathrm G(\Q) as the group of integer points \Gamma = \mathrm_n(\Z) \cap \mathrm G(\Q). In general it is not so obvious how to make precise sense of the notion of "integer points" of a \Q-group, and the subgroup defined above can change when we take different embeddings \mathrm G \to \mathrm_n(\Q). Thus a better notion is to take for definition of an arithmetic subgroup of \mathrm G(\Q) any group \Lambda which is commensurable (this means that both \Gamma/(\Gamma\cap \Lambda) and \Lambda/(\Gamma \cap \Lambda) are finite sets) with a group \Gamma defined as above (with respect to any embedding into \mathrm_n). With this definition, to the algebraic group \mathrm G is associated a collection of "discrete" subgroups all commensurable to each other.


Using number fields

A natural generalisation of the construction above is as follows: let F be a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
with ring of integers O and \mathrm G an algebraic group over F. If we are given an embedding \rho : \mathrm \to \mathrm_n defined over F then the subgroup \rho^(\mathrm_n(O)) \subset \mathrm G(F) can legitimately be called an arithmetic group. 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 \mathrm G' over \Q obtained by restricting scalars from F to \Q and the \Q -embedding \rho' : \mathrm G' \to \mathrm_ induced by \rho (where d = :\Q /math>) then the group constructed above is equal to (\rho')^(\mathrm_(\Z)).


Examples

The classical example of an arithmetic group is \mathrm_n(\Z), or the closely related groups \mathrm_n(\Z), \mathrm_n(\Z) and \mathrm_n(\Z). For n = 2 the group \mathrm_2(\Z), or sometimes \mathrm_2(\Z), is called the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
as it is related to the
modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular ...
. Similar examples are the Siegel modular groups \mathrm_(\Z). Other well-known and studied examples include the
Bianchi group In mathematics, a Bianchi group is a group of the form :PSL_2(\mathcal_d) where ''d'' is a positive square-free integer. Here, PSL denotes the projective special linear group and \mathcal_d is the ring of integers of the imaginary quadratic fiel ...
s \mathrm_2(O_), where m > 0 is a square-free integer and O_ is the ring of integers in the field \Q(\sqrt), and the Hilbert–Blumenthal modular groups \mathrm_2(O_m). Another classical example is given by the integral elements in the orthogonal group of a quadratic form defined over a number field, for example \mathrm(n,1)(\Z ). A related construction is by taking the unit groups of
orders Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
in
quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
s over number fields (for example the
Hurwitz quaternion order The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces. ...
). Similar constructions can be performed with unitary groups of
hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
s, a well-known example is the
Picard modular group In mathematics, a Picard modular group, studied by , is a group of the form SU(''J'',''L''), where ''L'' is a 3-dimensional lattice over the ring of integers of an imaginary quadratic field and ''J'' is a hermitian form on ''L'' of signature  ...
.


Arithmetic lattices in semisimple Lie groups

When G is a Lie group one can define an arithmetic lattice in G as follows: for any algebraic group \mathrm G defined over \Q such that there is a morphism \mathrm G(\R) \to G with compact kernel, the image of an arithmetic subgroup in \mathrm G(\Q) is an arithmetic lattice in G. Thus, for example, if G = \mathrm G(\R) and G is a subgroup of \mathrm_n then G \cap \mathrm_n(\Z) is an arithmetic lattice in G (but there are many more, corresponding to other embeddings); for instance, \mathrm_n(\Z) is an arithmetic lattice in \mathrm_n(\R ).


The Borel–Harish-Chandra theorem

A
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
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 Cocompact may refer to: * Cocompact group action * Cocompact Coxeter group * Cocompact embedding * Cocompact lattice {{dab ...
if and only if the "form" of G used to define it (i.e. the \Q -group \mathrm G) is anisotropic. For example, the arithmetic lattice associated to a quadratic form in n variables over \Q will be co-compact in the associated orthogonal group if and only if the quadratic form does not vanish at any point in \Q^n \setminus \.


Margulis arithmeticity theorem

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. For example, all lattices in \mathrm_n(\R ) are arithmetic when n \ge 3. The main new ingredient that Margulis used to prove his theorem was the superrigidity of lattices in higher-rank groups that he proved for this purpose. Irreducibility only plays a role when G has a factor of real rank one (otherwise the theorem always holds) and is not simple: it means that for any product decomposition G = G_1\times G_2 the lattice is not commensurable to a product of lattices in each of the factors G_i. For example, the lattice \mathrm_2(\Z sqrt 2 in \mathrm_2(\R) \times \mathrm_2(\R) is irreducible, while \mathrm_2(\Z) \times \mathrm_2(\Z) is not. The Margulis arithmeticity (and superrigidity) theorem holds for certain rank 1 Lie groups, namely \mathrm(n,1) for n \geqslant 1 and the exceptional group F_4^. It is known not to hold in all groups \mathrm(n,1) for n \geqslant 2 (ref to GPS) and for \mathrm(n, 1) when n = 1,2,3. There are no known non-arithmetic lattices in the groups \mathrm(n,1) when n \geqslant 4.


Arithmetic Fuchsian and Kleinian groups

An arithmetic Fuchsian group is constructed from the following data: a
totally real number field In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyn ...
F, a
quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
A over F and an order \mathcal O in A. It is asked that for one embedding \sigma: F \to \R the algebra A^\sigma \otimes_F \R be isomorphic to the matrix algebra M_2(\R) and for all others to the Hamilton quaternions. Then the group of units \mathcal O^1 is a lattice in (A^\sigma \otimes_F \R)^1 which is isomorphic to \mathrm_2(\R), and it is co-compact in all cases except when A is the matrix algebra over \Q. All arithmetic lattices in \mathrm_2(\R) are obtained in this way (up to commensurability). Arithmetic Kleinian groups are constructed similarly except that F is required to have exactly one complex place and A to be the Hamilton quaternions at all real places. They exhaust all arithmetic commensurability classes in \mathrm_2(\Complex).


Classification

For every semisimple Lie group G it is in theory possible to classify (up to commensurability) all arithmetic lattices in G, in a manner similar to the cases G = \mathrm_2(\R), \mathrm_2(\Complex) explained above. This amounts to classifying the algebraic groups whose real points are isomorphic up to a compact factor to G.


The congruence subgroup problem

A
congruence subgroup In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible matrix, invertible 2 × 2 integer matrices of determinan ...
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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
) 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).


S-arithmetic groups

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-arithmetic lattice'' (where S stands for the set of primes inverted). The prototypical example is \mathrm_2 \left( \Z \left tfrac 1 p \right\right). They are also naturally lattices in certain topological groups, for example \mathrm_2 \left( \Z \left tfrac 1 p \right\right) is a lattice in \mathrm_2(\R) \times \mathrm_2(\Q_p).


Definition

The formal definition of an S-arithmetic group for S a finite set of prime numbers is the same as for arithmetic groups with \mathrm_n(\Z) replaced by \mathrm_n\left(\Z \left \tfrac 1 N \right\right) where N is the product of the primes in S.


Lattices in Lie groups over local fields

The Borel–Harish-Chandra theorem generalizes to S-arithmetic groups as follows: if \Gamma is an S-arithmetic group in a \Q-algebraic group \mathrm G then \Gamma is a lattice in the
locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
:G = \mathrm G(\R) \times \prod_ \mathrm G(\Q_p).


Some applications


Explicit expander graphs

Arithmetic groups with
Kazhdan's property (T) In mathematics, a locally compact topological group ''G'' has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Spectrum of a C*-algebra, Fell topology. Informally, this means that if ''G'' acts un ...
or the weaker property (\tau) of Lubotzky and Zimmer can be used to construct expander graphs (Margulis), or even
Ramanujan graphs In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. AMurty's survey papernotes, Raman ...
(Lubotzky—Phillips—Sarnak). Such graphs are known to exist in abundance by probabilistic results but the explicit nature of these constructions makes them interesting.


Extremal surfaces and graphs

Congruence covers of arithmetic surfaces are known to give rise to surfaces with large
injectivity radius This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provi ...
. Likewise the Ramanujan graphs constructed by Lubotzky—Phillips—Sarnak have large
girth Girth may refer to: ;Mathematics * Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space * Girth (geometry), the perimeter of a parallel projection of a shape * Girth ...
. It is in fact known that the Ramanujan property itself implies that the local girths of the graph are almost always large.


Isospectral manifolds

Arithmetic groups can be used to construct isospectral manifolds. This was first realised by
Marie-France Vignéras Marie-France Vignéras (born 1946) is a French mathematician. She is a Professor Emeritus of the Institut de Mathématiques de Jussieu in Paris. She is known for her proof published in 1980 of the existence of isospectral non-isometric Riemann su ...
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.


Fake projective planes

A fake projective plane is a
complex surface Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
which has the same
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s as the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
\mathbb P^2(\Complex) but is not biholomorphic to it; the first example was discovered by Mumford. By work of Klingler (also proved independently by Yeung) all such are quotients of the 2-ball by arithmetic lattices in \mathrm(2,1). The possible lattices have been classified by Prasad and Yeung and the classification was completed by Cartwright and Steger who checked that they actually correspond to fake projective planes.


References

{{DEFAULTSORT:Arithmetic Group Algebraic groups Group theory Number theory Differential geometry