Envelopes of certain solvable groups

A discrete subgroup $\Gamma$ of a locally compact group $H$ is called a uniform lattice if the quotient $H/\Gamma$ is compact. Such an $H$ is called an envelope of $\Gamma$. In this paper we study the problem of classifying envelopes of various solvable groups including the solvable Baumslag-Solitar groups, lamplighter groups and certain abelian-by-cyclic groups. Our techniques are geometric and quasi-isometric in nature. In particular we show that for every $\Gamma$ we consider there is a finite family of preferred model spaces $X$ such that, up to compact groups, $H$ is a cocompact subgroup of $Isom(X)$.