Large facing tuples and a strengthened sector lemma

TL;DR

This paper strengthens the sector lemma for certain CAT(0) cube complexes, providing simplified proofs and collecting key results, with implications for cubical group theory and geometric rigidity.

Contribution

It introduces a strengthened sector lemma for hyperplane-essential CAT(0) cube complexes and offers simplified proofs of known results, enhancing understanding of cube complex geometry.

Findings

Every quarterspace contains a halfspace in hyperplane-essential complexes

Simplified proofs of loxodromic isometries of the contact graph

Collected results linking Ramsey's theorem and Dilworth's theorem in cube complexes

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 \emph{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.