Groups Quasi‐Isometric to H 2 × R

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 H 2 × R is a finite extension of a cocompact lattice in Isom( H 2 × R ) or Isom(S L ˜ (2, R )).