Rank Rigidity for Cat(0) Cube Complexes

Pierre Emmanuel Caprace; Michah Sageev

doi:https://doi.org/10.1007/s00039-011-0126-7

Rank Rigidity for Cat(0) Cube Complexes

Pierre Emmanuel Caprace, Michah Sageev

Mathematics

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	851-891
Number of pages	41
Journal	Geometric and Functional Analysis
Volume	21
Issue number	4
DOIs	https://doi.org/10.1007/s00039-011-0126-7
State	Published - Aug 2011

Keywords

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

All Science Journal Classification (ASJC) codes

Analysis
Geometry and Topology

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https://doi.org/10.1007/s00039-011-0126-7

Cite this

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title = "Rank Rigidity for Cat(0) Cube Complexes",

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.",

keywords = "CAT(0) space, Rank rigidity, Tits alternative, cube complex, rank-one isometry",

author = "Caprace, {Pierre Emmanuel} and Michah Sageev",

note = "Funding Information: M.S. is supported in part by ISF grant #580/07. Funding Information: Keywords and phrases: Rank rigidity, rank-one isometry, cube complex, CAT(0) space, Tits alternative 2010 Mathematics Subject Classification: 20F65, 20F67, 53C21 P.-E.C. is an F.R.S.-FNRS research associate, supported in part by FNRS grant F.4520.11.",

year = "2011",

month = aug,

doi = "https://doi.org/10.1007/s00039-011-0126-7",

language = "الإنجليزيّة",

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pages = "851--891",

journal = "Geometric and Functional Analysis",

issn = "1016-443X",

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TY - JOUR

T1 - Rank Rigidity for Cat(0) Cube Complexes

AU - Caprace, Pierre Emmanuel

AU - Sageev, Michah

N1 - Funding Information: M.S. is supported in part by ISF grant #580/07. Funding Information: Keywords and phrases: Rank rigidity, rank-one isometry, cube complex, CAT(0) space, Tits alternative 2010 Mathematics Subject Classification: 20F65, 20F67, 53C21 P.-E.C. is an F.R.S.-FNRS research associate, supported in part by FNRS grant F.4520.11.

PY - 2011/8

Y1 - 2011/8

N2 - 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.

AB - 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.

KW - CAT(0) space

KW - Rank rigidity

KW - Tits alternative

KW - cube complex

KW - rank-one isometry

UR - http://www.scopus.com/inward/record.url?scp=80051697167&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s00039-011-0126-7

DO - https://doi.org/10.1007/s00039-011-0126-7

M3 - مقالة

SN - 1016-443X

VL - 21

SP - 851

EP - 891

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 4

ER -

Rank Rigidity for Cat(0) Cube Complexes

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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