Envelopes of certain solvable groups

A discrete subgroup $Γ$ of a locally compact group $H$ is called a uniform lattice if the quotient $H/Γ$ is compact. Such an $H$ is called an envelope of $Γ$. 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 $Γ$ we consider there is a finite family of preferred model spaces $X$ such that, up to compact groups, $H$ is a cocompact subgroup of $Isom(X)$.

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