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Large facing tuples
and a strengthened sector lemma
Mark Hagen |
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Vol. 4 (2022), No. 1, 55–86
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DOI: 10.2140/tunis.2022.4.55
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Abstract |
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We prove a strengthened sector lemma for irreducible, finite-dimensional, locally finite, essential, cocompact CAT(0) cube complexes under the additional hypothesis that the complex is hyperplane-essential; we prove that every quarterspace contains a halfspace. In aid of this, we present simplified proofs of known results about loxodromic isometries of the contact graph, avoiding the use of disc diagrams. This paper has an expository element; in particular, we collect results about cube complexes proved by combining Ramsey’s theorem and Dilworth’s theorem. We illustrate the use of these tricks with a discussion of the Tits alternative for cubical groups, and ask some questions about “quantifying” statements related to rank rigidity and the Tits alternative. |
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Keywords
CAT(0) cube complex, contact graph, sector lemma, cubical
sector, halfspace, Ramsey's theorem, Tits alternative,
Dilworth's theorem
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Mathematical Subject Classification
Primary: 20F65
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Milestones
Received: 25 October 2020
Revised: 27 January 2021
Accepted: 11 February 2021
Published: 30 March 2022
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Authors |
| Mark Hagen | |
| School of Mathematics University of Bristol Fry Building Woodland Road Bristol BS8 1UG United Kingdom |
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