Uniform lattice in semidirect product A uniform lattice in a locally compact group $G$ is a discrete subgroup $\Gamma\subset G$ such that $G/\Gamma$ is compact.
My question is whether a uniform lattice exists in the group
$$
G={\mathbb R}^2\rtimes SL_2({\mathbb R}).
$$
This group is unimodular, so an obvious criterion is satisfied. It also admits a lattice (=discrete subgroup of finite covolume), for example the group of integer valued points ${\mathbb Z}^2\rtimes SL_2({\mathbb Z})$, but this is not cocompact.
 A: No, there's no uniform (=cocompact) lattice in $\mathbf{R}^n\rtimes\mathrm{SL}_n(\mathbf{R})$ for any $n\ge 2$. Up to the action by automorphisms of $\mathrm{GL}_n(\mathbf{R})$, all lattices are contained in $\mathbf{R}^n\rtimes\mathrm{SL}_n(\mathbf{Z})$, which is not cocompact.
Indeed, let $\Gamma$ be a lattice. In a connected Lie group, the intersection of a lattice with the amenable radical is a lattice in the amenable radical. So $\Gamma\cap\mathbf{R}^n$ is a lattice in $\mathbf{R}^n$; hence modulo a global automorphism induced by some element of $\mathrm{GL}_n(\mathbf{R})$, we can assume that $\Gamma\cap\mathbf{R}^n=\mathbf{Z}^n$. The subgroup $\Gamma$ is then contained in the normalizer of $\Gamma\cap\mathbf{R}^n=\mathbf{Z}^n$, and this normalizer is equal to $\mathbf{R}^n\rtimes\mathrm{SL}_n(\mathbf{Z})$.

(Edit; added to reflect the comments:) On the other hand, $\mathbf{R}^3\rtimes\mathrm{SL}_2(\mathbf{R})$ admits cocompact lattices. Indeed, let $q=X^2+Y^2-7Z^2$. Let $G(q)(K)=K^3\rtimes\mathrm{SO}(q)(K)$. Then $q$ is $\mathbf{Q}$-anisotropic; so $G(\mathbf{Z})$ is a cocompact lattice in $G(\mathbf{R})=\mathbf{R}^3\rtimes \mathrm{SO}(q)(\mathbf{R})$. Pulling back, we get a cocompact lattice in $\mathbf{R}^3\rtimes \mathrm{SL}_2(\mathbf{R})$.
A: I'm a bit late to the party, but just want to remark that the reference given by YCor (Corollary 8.28 in Raghunathan) is actually known to be incorrect.
Counterexamples are known, first due to Starkov (1984).
Luckily, the statement is still correct if the Levi subgroup of G has no compact factors, which is the case here (see e.g. Starkov's book "Dynamical systems on homogeneous spaces", Section E).
(Apologies for not using the "comment" function, but I don't have permission to write comments yet.)
