Uniform subgroup
of a locally compact topological group
A closed subgroup such that the quotient space
is compact. Closely related to this notion is that of a quasi-uniform subgroup of
, that is, a closed subgroup
of
for which there is a
-invariant measure
on
with
. For example, the subgroup
of
is quasi-uniform, but not uniform. On the other hand, the subgroup
of all upper-triangular matrices in
is a uniform subgroup of
that is not quasi-uniform (there are no
-invariant measures on
). However, every connected quasi-uniform subgroup of a Lie group
is a uniform subgroup (see [1]), and every discrete uniform subgroup of
is quasi-uniform [2]. (On the topic of discrete uniform subgroups of Lie groups, see Discrete subgroup.) If
is a connected Lie group and
is a uniform subgroup of
, then the normalizer
in
of the connected component of the identity
in
contains a maximal connected triangular subgroup of
(see [3]). An algebraic subgroup
of a connected algebraic complex linear Lie group
is a uniform subgroup if and only if
is a parabolic subgroup in
. All connected uniform subgroups of semi-simple Lie groups have been described (see [4]). A non-discrete uniform subgroup
of a connected semi-simple Lie group
has the property of strong rigidity (see [5]), which is that in
there are a finite number of subgroups
,
, such that any subgroup
isomorphic to
is conjugate to one of the subgroups
. Important examples of uniform and quasi-uniform subgroups are constructed as follows. Let
be a linear algebraic group defined over the field of rational numbers
, let
be the adèle group and let
be the subgroup of principal adèles. Then
is a discrete subgroup in
; moreover,
is a uniform subgroup of
if and only if: 1)
has no non-trivial rational characters defined over
; and 2) all unipotent elements of
belong to its radical (see [6], [7]). In particular, if
is a unipotent algebraic group defined over
, then
is a uniform subgroup of
. Condition 1) is necessary and sufficient for the quasi-uniformity of
and
.
References
[1] | G.D. Mostow, "Homogeneous spaces with finite invariant measure" Ann. of Math. , 75 : 1 (1962) pp. 17–37 |
[2] | M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) |
[3] | A.L. Onishchik, "Lie groups transitive on compact manifolds" Transl. Amer. Math. Soc. (2) , 73 (1968) pp. 59–72 Mat. Sb. , 71 : 4 (1966) pp. 483–494 |
[4] | A.L. Onishchik, "On Lie groups transitive on compact manifolds II" Math. USSR Sb. , 3 : 3 (1967) pp. 373–388 Mat. Sb. , 74 : 3 (1967) pp. 398–416 |
[5] | M. Goto, H.C. Wang, "Non-discrete uniform subgroups of semisimple Lie groups" Math. Ann. , 198 : 4 (1972) pp. 259–286 |
[6] | A. Borel, "Some properties of adele groups attached to algebraic groups" Bull. Amer. Math. Soc. , 67 : 6 (1961) pp. 583–585 |
[7] | G.D. Mostow, T. Tamagawa, "On the compactness of arithmetically defined homogeneous spaces" Ann. of Math. , 76 : 3 (1962) pp. 446–463 |
Uniform subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_subgroup&oldid=15977