Arithmetic group
A subgroup of a linear algebraic group
defined over the field
of rational numbers, that satisfies the following condition: There exists a faithful rational representation
defined over
(cf. Representation theory) such that
is commensurable with
, where
is the ring of integers (two subgroups
and
of a group
are called commensurable if
is of finite index in
and in
). This condition is then also satisfied for any other faithful representation defined over
. More generally, an arithmetic group is a subgroup of an algebraic group
, defined over a global field
, that is commensurable with the group
of
-points of
, where
is the ring of integers of
. An arithmetic group
is a discrete subgroup of
.
If is a
-epimorphism of algebraic groups, then the image
of any arithmetic group
is an arithmetic group in
[1]. The name arithmetic group is sometimes also given to an abstract group that is isomorphic to an arithmetic subgroup of some algebraic group. Thus, if
is an algebraic number field, the group
, where
is obtained from
by restricting the field of definition from
to
, is called an arithmetic group. In the theory of Lie groups the name arithmetic subgroups is also given to images of arithmetic subgroups of the group of real points of
under the factorization of
by compact normal subgroups.
References
[1] | A. Borel, "Ensembles fundamentaux pour les groups arithmétiques et formes automorphes" , Fac. Sci. Paris (1967) |
[2] | A. Borel, Harish-Chandra, "Arithmetic subgroups of algebraic groups" Ann. of Math. , 75 (1962) pp. 485–535 |
[3] | , Arithmetic groups and discontinuous subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) |
Comments
Useful additional references are [a1]–[a3]. [a2] is an elementary introduction to the theory of arithmetic groups.
Conjectures of A. Selberg and I.I. Pyatetskii-Shapiro roughly state that for most semi-simple Lie groups discrete subgroups of finite co-volume are necessarily arithmetic. G.A. Margulis settled this question completely and, in particular, proved the conjectures in question. See Discrete subgroup for more detail.
References
[a1] | A. Borel, "Arithmetic properties of linear algebraic groups" , Proc. Internat. Congress mathematicians (Stockholm, 1962) , Inst. Mittag-Leffler (1963) pp. 10–22 |
[a2] | A. Borel, "Introduction aux groupes arithmétiques" , Hermann (1969) |
[a3] | J.E. Humphreys, "Arithmetic groups" , Springer (1980) |
Arithmetic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_group&oldid=21815