Uniform subgroup

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