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Polynomial growth of coarse intervals in coarse median spaces

Polynomial growth of coarse intervals in coarse median spaces
Polynomial growth of coarse intervals in coarse median spaces
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.
University of Southampton
Ladjali, Amina Assouda
9f086d63-e31f-4d7b-80b3-abbfda35787a
Ladjali, Amina Assouda
9f086d63-e31f-4d7b-80b3-abbfda35787a
Wright, Nick
f4685b8d-7496-47dc-95f0-aba3f70fbccd
Niblo, Graham
43fe9561-c483-4cdf-bee5-0de388b78944

Ladjali, Amina Assouda (2025) Polynomial growth of coarse intervals in coarse median spaces. University of Southampton, Doctoral Thesis, 121pp.

Record type: Thesis (Doctoral)

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.

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Published date: 13 April 2025

Identifiers

Local EPrints ID: 500707
URI: http://eprints.soton.ac.uk/id/eprint/500707
PURE UUID: dee3399c-ea86-4112-8370-e691f09f02c6
ORCID for Amina Assouda Ladjali: ORCID iD orcid.org/0000-0001-5302-1591
ORCID for Nick Wright: ORCID iD orcid.org/0000-0003-4884-2576
ORCID for Graham Niblo: ORCID iD orcid.org/0000-0003-0648-7027

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Date deposited: 12 May 2025 16:31
Last modified: 11 Sep 2025 03:10

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Contributors

Author: Amina Assouda Ladjali ORCID iD
Thesis advisor: Nick Wright ORCID iD
Thesis advisor: Graham Niblo ORCID iD

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