Inventiones mathematicae

, Volume 128, Issue 2, pp 393-416

Quasi-isometries preserve the geometric decomposition of Haken manifolds

  • Michael KapovichAffiliated withDepartment of Mathematics, University of Utah, Salt Lake City, UT 84112, USA (e-mail: kapovich@math.utah.edu)
  • , Bernhard LeebAffiliated withMathematisches Institut, Universität Bonn, Beringstr. 1, D-53115 Bonn, Germany (e-mail: leeb@rhein.iam.uni-bonn.de)

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Abstract.

We prove quasi-isometry invariance of the canonical decomposition for fundamental groups of Haken 3-manifolds with zero Euler characteristic. We show that groups quasi-isometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a by-product we describe all 2-dimensional quasi-flats in the universal covers of non-geometric Haken manifolds.