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.