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Commentarii Mathematici Helvetici

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Volume 90, Issue 1, 2015, pp. 195–224
DOI: 10.4171/CMH/351

Published online: 2015-02-23

Envelopes of certain solvable groups

Tullia Dymarz[1]

(1) University of Wisconsin, Madison, USA

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 $Isom(X)$. We also answer problem 10.4 in \cite{FM3} for a large class of abelian-by-cyclic groups.

Keywords: Lattices in locally compact groups, lamplighter groups, Baumslag–Solitar groups, quasi-isometries

Dymarz Tullia: Envelopes of certain solvable groups. Comment. Math. Helv. 90 (2015), 195-224. doi: 10.4171/CMH/351