Rank Rigidity for Cat(0) Cube Complexes

  • Pierre-Emmanuel CapraceEmail author
  • Michah Sageev


We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rankone isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.

Keywords and phrases

Rank rigidity rank-one isometry cube complex CAT(0) space Tits alternative 

2010 Mathematics Subject Classification

20F65 20F67 53C21 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.UCLouvain – IRMPLouvainla-NeuveBelgium
  2. 2.Department of MathematicsTechnionHaifaIsrael

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