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
Abstract

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.

Keywords
CAT(0) cube complex, contact graph, sector lemma, cubical sector, halfspace, Ramsey's theorem, Tits alternative, Dilworth's theorem
Mathematical Subject Classification
Primary: 20F65
Milestones
Received: 25 October 2020
Revised: 27 January 2021
Accepted: 11 February 2021
Published: 30 March 2022
Authors
Mark Hagen
School of Mathematics
University of Bristol
Fry Building
Woodland Road
Bristol
BS8 1UG
United Kingdom