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Difference between revisions of "Uniform subgroup"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.D. Mostow,   "Homogeneous spaces with finite invariant measure" ''Ann. of Math.'' , '''75''' : 1 (1962) pp. 17–37</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.S. Raghunathan,   "Discrete subgroups of Lie groups" , Springer (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Goto,   H.C. Wang,   "Non-discrete uniform subgroups of semisimple Lie groups" ''Math. Ann.'' , '''198''' : 4 (1972) pp. 259–286</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A. Borel,   "Some properties of adele groups attached to algebraic groups" ''Bull. Amer. Math. Soc.'' , '''67''' : 6 (1961) pp. 583–585</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> G.D. Mostow,   T. Tamagawa,   "On the compactness of arithmetically defined homogeneous spaces" ''Ann. of Math.'' , '''76''' : 3 (1962) pp. 446–463</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.D. Mostow, "Homogeneous spaces with finite invariant measure" ''Ann. of Math.'' , '''75''' : 1 (1962) pp. 17–37 {{MR|0145007}} {{ZBL|0115.25702}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) {{MR|0507234}} {{MR|0507236}} {{ZBL|0254.22005}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|0198.29001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|0198.28903}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Goto, H.C. Wang, "Non-discrete uniform subgroups of semisimple Lie groups" ''Math. Ann.'' , '''198''' : 4 (1972) pp. 259–286 {{MR|0354934}} {{ZBL|0228.22014}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A. Borel, "Some properties of adele groups attached to algebraic groups" ''Bull. Amer. Math. Soc.'' , '''67''' : 6 (1961) pp. 583–585 {{MR|0141671}} {{ZBL|0119.37002}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> G.D. Mostow, T. Tamagawa, "On the compactness of arithmetically defined homogeneous spaces" ''Ann. of Math.'' , '''76''' : 3 (1962) pp. 446–463 {{MR|0141672}} {{ZBL|0196.53201}} </TD></TR></table>

Revision as of 14:52, 24 March 2012

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 MR0145007 Zbl 0115.25702
[2] M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) MR0507234 MR0507236 Zbl 0254.22005
[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 Zbl 0198.29001
[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 Zbl 0198.28903
[5] M. Goto, H.C. Wang, "Non-discrete uniform subgroups of semisimple Lie groups" Math. Ann. , 198 : 4 (1972) pp. 259–286 MR0354934 Zbl 0228.22014
[6] A. Borel, "Some properties of adele groups attached to algebraic groups" Bull. Amer. Math. Soc. , 67 : 6 (1961) pp. 583–585 MR0141671 Zbl 0119.37002
[7] G.D. Mostow, T. Tamagawa, "On the compactness of arithmetically defined homogeneous spaces" Ann. of Math. , 76 : 3 (1962) pp. 446–463 MR0141672 Zbl 0196.53201
How to Cite This Entry:
Uniform subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_subgroup&oldid=21957
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article