Translator Disclaimer
2022 Large facing tuples and a strengthened sector lemma
Mark Hagen
Tunisian J. Math. 4(1): 55-86 (2022). 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.

Citation

Download Citation

Mark Hagen. "Large facing tuples and a strengthened sector lemma." Tunisian J. Math. 4 (1) 55 - 86, 2022. https://doi.org/10.2140/tunis.2022.4.55

Information

Received: 25 October 2020; Revised: 27 January 2021; Accepted: 11 February 2021; Published: 2022
First available in Project Euclid: 9 May 2022

Digital Object Identifier: 10.2140/tunis.2022.4.55

Subjects:
Primary: 20F65

Keywords: CAT(0) cube complex , contact graph , cubical sector , Dilworth’s theorem , halfspace , Ramsey’s theorem , sector lemma , Tits alternative

Rights: Copyright © 2022 Mathematical Sciences Publishers

JOURNAL ARTICLE
32 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.4 • No. 1 • 2022
MSP
Back to Top