Examples of homogeneous Riemannian manifolds not quasi-isometric to any finitely generated group (Q2738862)

As it is well known, a homogeneous Riemannian manifold \(X\) is isometric to a quotient space \(G/K\), where \(G\) is a transitive group of isometries of \(X\) and \(K\) is the stabilizer of a point of \(X\). \(K\) being compact, \(X\) is quasi isometric to \(G\). In general, \(X\) is not quasi isometric to a finitely generated group. NEWLINENEWLINENEWLINEThe aim of the paper under review is to characterize the nilpotent graded Lie groups which are quasi isometric to finitely generated groups. It is proved that a 1-connected nilpotent graded Lie group \(N\) is quasi isometric to a finitely generated group \(\Gamma\) if and only if \(N\) contains lattices.