Rank Rigidity for Cat(0) Cube Complexes

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...

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Published inGeometric and functional analysis Vol. 21; no. 4; pp. 851 - 891
Main Authors Caprace, Pierre-Emmanuel, Sageev, Michah
Format Journal Article
LanguageEnglish
Published Basel SP Birkhäuser Verlag Basel 01.08.2011
Subjects
Analysis
Mathematics
Mathematics and Statistics
20F65
Tits alternative
cube complex
53C21
rank-one isometry
CAT space
Rank rigidity
20F67
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Summary: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.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-011-0126-7