The quasi-isometry classification of lattices in semisimple Lie groups
This is a report on the recently completed quasi-isometry classification of lattices in semisimple Lie groups. A quasi-isometry between metric spaces is a map \(f:X\to Y\) such that \(d(f(x), f(y))\) differs in linear bounds from \(d(x,y)\). A lattice in a Lie group \(G\) becomes a metric space by some word metric. Two lattices \(\Gamma,\Gamma'\) are called commensurable in \(G\) if there is a \(g\in G\) such that \(\Gamma \cap g \Gamma'g^{-1}\) is a lattice. The following statements are easy to prove: (1) If \(\Gamma, \Gamma'\) are commensurable then they are quasi-isometric. (2) If \(\Gamma\) is uniform, i.e. \(G/\Gamma\) is compact, then \(\Gamma\) is quasi-isometric to \(\Gamma'\) if and only if \(\Gamma'\) also is uniform. It therefore remains to settle the classification up to quasi-isometry for nonuniform lattices. Clearly it suffices to consider irreducible lattices. The main result of the paper is that if the Lie group \(G\) is not locally isomorphic to \(SL_2 (\mathbb{R})\) then \(\Gamma,\Gamma'\) irreducible and quasi-isometric already implies that they are commensurable. In the case of \(SL_2 (\mathbb{R})\) on the contrary any two nonuniform lattices are quasi-isometric.
scientific article; zbMATH DE number 1117492