In order to have well defined geometries, finitely generated groups are studied up to quasi-isometric equivalence. This leads one to ask: Given a finitely generated group which other groups are quasi-isometric to it? Or more generally: Given a metric space  X which groups are quasi-isometric to X? Answering such questions gives quasi-isometric rigidity results. In these lectures we will survey techniques/results used to prove quasi-isometric rigidity theorems and then we will study more carefully the case when X  is a solvable Lie group with a left invariant metric

In order to have well defined geometries, finitely generated groups are studied up to quasi-isometric equivalence. This leads one to ask: Given a finitely generated group which other groups are quasi-isometric to it? Or more generally: Given a metric space  X which groups are quasi-isometric to X? Answering such questions gives quasi-isometric rigidity results. In these lectures we will survey techniques/results used to prove quasi-isometric rigidity theorems and then we will study more carefully the case when X  is a solvable Lie group with a left invariant metric.