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)).