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The most powerful geometric tools are those of differential geometry, but to apply such techniques to finitely generated groups seems hopeless at first glance since the natural metric on a finitely generated group is discrete. However Gromov recognized that a group can metrically resemble a manifold in such a way that geometric results about that manifold carry over to the group [18, 20]. This resemblance is formalized in the concept of a 'quasi-isometry'. This paper contributes to an ongoing program to understand which groups are quasi-isometric to which simply connected, homogeneous, Riemannian manifolds [15, 18, 20] by proving that any group quasi-isometric to H2×R is a finite extension of a cocompact lattice in Isom(H2×R) or Isom(SL...(2, R)). [ABSTRACT FROM PUBLISHER]