# 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 $G_A$.