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 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.
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P.-E.C. is an F.R.S.-FNRS research associate, supported in part by FNRS grant F.4520.11. M.S. is supported in part by ISF grant #580/07.
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Caprace, PE., Sageev, M. Rank Rigidity for Cat(0) Cube Complexes. Geom. Funct. Anal. 21, 851–891 (2011). https://doi.org/10.1007/s00039-011-0126-7
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DOI: https://doi.org/10.1007/s00039-011-0126-7