Vol. 4, No. 1, 2022

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Large facing tuples and a strengthened sector lemma

Mark Hagen

Vol. 4 (2022), No. 1, 55–86
DOI: 10.2140/tunis.2022.4.55

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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CAT(0) cube complex, contact graph, sector lemma, cubical sector, halfspace, Ramsey's theorem, Tits alternative, Dilworth's theorem
Mathematical Subject Classification
Primary: 20F65
Received: 25 October 2020
Revised: 27 January 2021
Accepted: 11 February 2021
Published: 30 March 2022
Mark Hagen
School of Mathematics
University of Bristol
Fry Building
Woodland Road
United Kingdom