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arXiv · 1005.5687

Rank rigidity for CAT(0) cube complexes

Pierre-Emmanuel Caprace·Michah Sageev
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Abstract

We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rank one 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.

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Pierre-Emmanuel Caprace, Michah Sageev. 2011-04-01. Rank rigidity for CAT(0) cube complexes. https://doi.org/10.1007/s00039-011-0126-7

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