View source for Uniform subgroup

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''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 [[#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====
<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>