The quasi-isometry classification of rank one lattices. (English) Zbl 0852.22010

The author answers Gersten’s question “When are two nonuniform lattices quasi-isometric to each other?” in the case of rank one semisimple Lie groups (such groups agree, up to index 2, with isometry groups of negatively curved symmetric spaces). Their lattices are rank one lattices. Let \(G\) be a Lie group, let \(L_1\), \(L_2 \subseteq G\) be lattices, then an element \(c \in G\) commensurates \(L_1\) to \(L_2\) if \(c \cdot L_1 \cdot c^{-1} \cap L_2\) has finite index in \(L_2\). Main Theorem: Let \(G\) be a rank one Lie group and let \(G\) be not the isometry group of the hyperbolic planes and let \(L_1\), \(L_2\) be nonuniform lattices in \(G\). Any quasi-isometry between \(L_1\) and \(L_2\) is equivalent to (the restriction of) an element of \(G\) which commensurates \(L_1\) to \(L_2\). – Corollaries: (1) Let \(\Gamma\) be a finitely generated group which is quasi-isometric to a nonuniform lattice in \(G\). Then \(\Gamma\) is a finite extension of a nonuniform lattice in \(G\).
(2) A nonuniform lattice in \(G\) is arithmetic iff it has infinite index in its quasi-isometry group.
Reviewer: B.F.Šmarda (Brno)


22E40 Discrete subgroups of Lie groups
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