Polynomial growth of coarse intervals in coarse median spaces

Abstract

In this thesis, we explore the structure and geometry of coarse intervals in coarse median spaces, obtaining polynomial growth of coarse intervals as a consequence. We study finite rank bounded geometry coarse intervals in a quasi-geodesic coarse median space. We equip our coarse intervals with an ordering and then split our approach: we first consider rank 2 intervals and then turn our attention to higher rank intervals. In the rank 2 case, we introduce the concept of a coarse hyperplane in a coarse median space, a coarse analogue of hyperplanes in CAT(0) cube complexes, and use this as an important tool in proving three key properties: coarse hyperplanes intersected with a rank 2 coarse interval have co-dimension 1 in the coarse interval, coarse hyperplanes coarsely cover the whole coarse interval, and the intersection of a coarse hyperplane and coarse interval is ‘almost’ a coarse interval. We then use these three results and an inductive argument to show that rank 2 coarse intervals have quadratic growth. For the higher rank case, we introduce the important notion of a directly edge maximal point in a coarsely convex subset of a coarse interval. We then show that the length of a finite, incomparable sequence of directly edge maximal points, an antichain, associated to a coarsely convex subset is bounded above by the rank of the subset. Equipped with this result, we prove that an R-separated subset of directly edge maximal points equipped with a partial ordering can be decomposed into a union of chains via the aforementioned result and the application of Dilworth’s Lemma. We then obtain two maps, ƒ= ( ƒi) and g, where f maps any point u in a coarse interval to a product of chains, which is isometrically embedded in Ζr. Each chain gives the coordinate of u in that direction, i.e. ƒi provides the ith coordinate of u. The map g maps the coordinates of u back into the interval by computing the minimum of these coordinates. Hence, we have shown that higher rank coarse intervals also have polynomial growth.

More information

Identifiers

ORCID for Amina Assouda Ladjali: orcid.org/0000-0001-5302-1591

ORCID for Nick Wright: orcid.org/0000-0003-4884-2576

ORCID for Graham Niblo: orcid.org/0000-0003-0648-7027

Catalogue record

Export record