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GROUPS QUASI-ISOMETRIC TO H2×R

Published online by Cambridge University Press:  24 August 2001

ELEANOR G. RIEFFEL
ELEANOR G. RIEFFEL
Affiliation:
FX Palo Alto Laboratory, Building 4, 3400 Hillview Avenue, Palo Alto, CA 94304, USA; [email protected]
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Abstract

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)).

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 64 , Issue 1 , August 2001 , pp. 44 - 60
Copyright
The London Mathematical Society 2001

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