Rank rigidity for CAT(0) cube complexes

Details Rank rigidity for CAT(0) cube complexes Rank rigidity for CAT(0) cube complexes

2010-05-31

arxiv.org/abs/1005.5687v3

Geom. Funct. Anal. 21 Nr. 4 (2011) pp. 851--891

Mathematics

Group Theory

39 pages, 4 figures. Revised version according to referee report

Scientific paper

10.1007/s00039-011-0126-7

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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