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Groups Quasi‐Isometric to H 2 × R
Author(s) -
Rieffel Eleanor G.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1017/s0024610701002034
Subject(s) - isometry group , mathematics , group (periodic table) , isometric exercise , pure mathematics , manifold (fluid mechanics) , isometry (riemannian geometry) , discrete group , riemannian manifold , differential geometry , simply connected space , combinatorics , homogeneous , differential (mechanical device) , physics , medicine , mechanical engineering , quantum mechanics , engineering , physical therapy , thermodynamics
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 )).