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