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''of a locally compact topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u0952701.png" />''
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''of a locally compact topological group $G$''
  
A closed subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u0952702.png" /> such that the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u0952703.png" /> is compact. Closely related to this notion is that of a quasi-uniform subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u0952704.png" />, that is, a closed subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u0952705.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u0952706.png" /> for which there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u0952707.png" />-invariant measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u0952708.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u0952709.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527010.png" />. For example, the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527012.png" /> is quasi-uniform, but not uniform. On the other hand, the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527013.png" /> of all upper-triangular matrices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527014.png" /> is a uniform subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527015.png" /> that is not quasi-uniform (there are no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527016.png" />-invariant measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527017.png" />). However, every connected quasi-uniform subgroup of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527018.png" /> is a uniform subgroup (see [[#References|[1]]]), and every discrete uniform subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527019.png" /> is quasi-uniform [[#References|[2]]]. (On the topic of discrete uniform subgroups of Lie groups, see [[Discrete subgroup|Discrete subgroup]].) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527020.png" /> is a connected [[Lie group|Lie group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527021.png" /> is a uniform subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527022.png" />, then the normalizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527024.png" /> of the connected component of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527026.png" /> contains a maximal connected triangular subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527027.png" /> (see [[#References|[3]]]). An algebraic subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527028.png" /> of a connected algebraic complex linear Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527029.png" /> is a uniform subgroup if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527030.png" /> is a [[Parabolic subgroup|parabolic subgroup]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527031.png" />. All connected uniform subgroups of semi-simple Lie groups have been described (see [[#References|[4]]]). A non-discrete uniform subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527032.png" /> of a connected semi-simple Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527033.png" /> has the property of strong rigidity (see [[#References|[5]]]), which is that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527034.png" /> there are a finite number of subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527036.png" />, such that any subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527037.png" /> isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527038.png" /> is conjugate to one of the subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527039.png" />. Important examples of uniform and quasi-uniform subgroups are constructed as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527040.png" /> be a [[Linear algebraic group|linear algebraic group]] defined over the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527041.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527042.png" /> be the adèle group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527043.png" /> be the subgroup of principal adèles. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527044.png" /> is a discrete subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527045.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527046.png" /> is a uniform subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527047.png" /> if and only if: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527048.png" /> has no non-trivial rational characters defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527049.png" />; and 2) all unipotent elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527050.png" /> belong to its radical (see [[#References|[6]]], [[#References|[7]]]). In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527051.png" /> is a unipotent algebraic group defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527052.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527053.png" /> is a uniform subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527054.png" />. Condition 1) is necessary and sufficient for the quasi-uniformity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095270/u09527056.png" />.
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A closed subgroup $H\subset G$ such that the quotient space $G/H$ is compact. Closely related to this notion is that of a quasi-uniform subgroup of $G$, that is, a closed subgroup $H$ of $G$ for which there is a $G$-invariant measure $\mu$ on $G/H$ with $\mu(G/H)<\infty$. For example, the subgroup $\SL_2(\mathbf Z)$ of $\SL_2(\mathbf R)$ is quasi-uniform, but not uniform. On the other hand, the subgroup $T$ of all upper-triangular matrices in $\SL_2(\mathbf R)$ is a uniform subgroup of $\SL_2(\mathbf R)$ that is not quasi-uniform (there are no $\SL_2(\mathbf R)$-invariant measures on $\SL_2(\mathbf R)/T$). However, every connected quasi-uniform subgroup of a Lie group $G$ is a uniform subgroup (see [[#References|[1]]]), and every discrete uniform subgroup of $G$ is quasi-uniform [[#References|[2]]]. (On the topic of discrete uniform subgroups of Lie groups, see [[Discrete subgroup|Discrete subgroup]].) If $G$ is a connected [[Lie group|Lie group]] and $H$ is a uniform subgroup of $G$, then the normalizer $N_G(H^0)$ in $G$ of the connected component of the identity $H^0$ in $H$ contains a maximal connected triangular subgroup of $G$ (see [[#References|[3]]]). An algebraic subgroup $H$ of a connected algebraic complex linear Lie group $G$ is a uniform subgroup if and only if $H$ is a [[Parabolic subgroup|parabolic subgroup]] in $G$. All connected uniform subgroups of semi-simple Lie groups have been described (see [[#References|[4]]]). A non-discrete uniform subgroup $H$ of a connected semi-simple Lie group $G$ has the property of strong rigidity (see [[#References|[5]]]), which is that in $G$ there are a finite number of subgroups $H_i$, $i=1,\dots,m$, such that any subgroup $H'\subset G$ isomorphic to $H$ is conjugate to one of the subgroups $H_i$. Important examples of uniform and quasi-uniform subgroups are constructed as follows. Let $G$ be a [[Linear algebraic group|linear algebraic group]] defined over the field of rational numbers $\mathbf Q$, let $G_A$ be the adèle group and let $G_{\mathbf Q}\subset G_A$ be the subgroup of principal adèles. Then $G_{\mathbf Q}$ is a discrete subgroup in $G_A$; moreover, $G_{\mathbf Q}$ is a uniform subgroup of $G_A$ if and only if: 1) $G$ has no non-trivial rational characters defined over $\mathbf Q$; and 2) all unipotent elements of $G_{\mathbf Q}$ belong to its radical (see [[#References|[6]]], [[#References|[7]]]). In particular, if $G$ is a unipotent algebraic group defined over $\mathbf Q$, then $G_{\mathbf Q}$ is a uniform subgroup of $G_A$. Condition 1) is necessary and sufficient for the quasi-uniformity of $G_{\mathbf Q}$ and $G_A$.
  
 
====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>

Latest revision as of 12:18, 30 December 2018

of a locally compact topological group $G$

A closed subgroup $H\subset G$ such that the quotient space $G/H$ is compact. Closely related to this notion is that of a quasi-uniform subgroup of $G$, that is, a closed subgroup $H$ of $G$ for which there is a $G$-invariant measure $\mu$ on $G/H$ with $\mu(G/H)<\infty$. For example, the subgroup $\SL_2(\mathbf Z)$ of $\SL_2(\mathbf R)$ is quasi-uniform, but not uniform. On the other hand, the subgroup $T$ of all upper-triangular matrices in $\SL_2(\mathbf R)$ is a uniform subgroup of $\SL_2(\mathbf R)$ that is not quasi-uniform (there are no $\SL_2(\mathbf R)$-invariant measures on $\SL_2(\mathbf R)/T$). However, every connected quasi-uniform subgroup of a Lie group $G$ is a uniform subgroup (see [1]), and every discrete uniform subgroup of $G$ is quasi-uniform [2]. (On the topic of discrete uniform subgroups of Lie groups, see Discrete subgroup.) If $G$ is a connected Lie group and $H$ is a uniform subgroup of $G$, then the normalizer $N_G(H^0)$ in $G$ of the connected component of the identity $H^0$ in $H$ contains a maximal connected triangular subgroup of $G$ (see [3]). An algebraic subgroup $H$ of a connected algebraic complex linear Lie group $G$ is a uniform subgroup if and only if $H$ is a parabolic subgroup in $G$. All connected uniform subgroups of semi-simple Lie groups have been described (see [4]). A non-discrete uniform subgroup $H$ of a connected semi-simple Lie group $G$ has the property of strong rigidity (see [5]), which is that in $G$ there are a finite number of subgroups $H_i$, $i=1,\dots,m$, such that any subgroup $H'\subset G$ isomorphic to $H$ is conjugate to one of the subgroups $H_i$. Important examples of uniform and quasi-uniform subgroups are constructed as follows. Let $G$ be a linear algebraic group defined over the field of rational numbers $\mathbf Q$, let $G_A$ be the adèle group and let $G_{\mathbf Q}\subset G_A$ be the subgroup of principal adèles. Then $G_{\mathbf Q}$ is a discrete subgroup in $G_A$; moreover, $G_{\mathbf Q}$ is a uniform subgroup of $G_A$ if and only if: 1) $G$ has no non-trivial rational characters defined over $\mathbf Q$; and 2) all unipotent elements of $G_{\mathbf Q}$ belong to its radical (see [6], [7]). In particular, if $G$ is a unipotent algebraic group defined over $\mathbf Q$, then $G_{\mathbf Q}$ is a uniform subgroup of $G_A$. Condition 1) is necessary and sufficient for the quasi-uniformity of $G_{\mathbf Q}$ and $G_A$.

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=15977
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article