Arithmetic group
Encyclopedia
In mathematics
, an arithmetic group (arithmetic subgroup) in a linear algebraic group
G defined over a number field K is a subgroup
Γ of G(K) that is commensurable
with G(O), where O is the ring of integers
of K. Here two subgroups A and B of a group are commensurable when their intersection
has finite index
in each of them. It can be shown that this condition depends only on G, not on a given matrix representation of G.
Examples of arithmetic groups include therefore the groups GLn(Z). The idea of arithmetic group is closely related to that of lattice
in a Lie group
. Lattices in that sense tend to be arithmetic, except in well-defined circumstances. The exact relationship of the two concepts was established by the work of Margulis
on superrigidity
. The general theory of arithmetic groups was developed by Armand Borel
and Harish-Chandra
; the description of their fundamental domain
s was in classical terms the reduction theory of algebraic forms.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an arithmetic group (arithmetic subgroup) in a linear algebraic group
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
G defined over a number field K is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
Γ of G(K) that is commensurable
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.-History of the concept:...
with G(O), where O is the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
of K. Here two subgroups A and B of a group are commensurable when their intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
has finite index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
in each of them. It can be shown that this condition depends only on G, not on a given matrix representation of G.
Examples of arithmetic groups include therefore the groups GLn(Z). The idea of arithmetic group is closely related to that of lattice
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
in a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
. Lattices in that sense tend to be arithmetic, except in well-defined circumstances. The exact relationship of the two concepts was established by the work of Margulis
Grigory Margulis
Gregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...
on superrigidity
Superrigidity
In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group G can, under some circumstances, be as good as a representation of G itself...
. The general theory of arithmetic groups was developed by Armand Borel
Armand Borel
Armand Borel 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...
and Harish-Chandra
Harish-Chandra
Harish-Chandra was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups. -Life:...
; the description of their fundamental domain
Fundamental domain
In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...
s was in classical terms the reduction theory of algebraic forms.