Coarse median algebras: The intrinsic geometry of coarse median spaces and their intervals
Coarse median algebras: The intrinsic geometry of coarse median spaces and their intervals
This paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of δ-hyperbolicity.
Coarse interval structures and a canonical metric, Coarse median spaces, Median algebras
Niblo, Graham
43fe9561-c483-4cdf-bee5-0de388b78944
Wright, Nicholas
f4685b8d-7496-47dc-95f0-aba3f70fbccd
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
May 2021
Niblo, Graham
43fe9561-c483-4cdf-bee5-0de388b78944
Wright, Nicholas
f4685b8d-7496-47dc-95f0-aba3f70fbccd
Zhang, Jiawen
aa149f14-dd1d-42b0-b863-623d1fedd1f5
Niblo, Graham, Wright, Nicholas and Zhang, Jiawen
(2021)
Coarse median algebras: The intrinsic geometry of coarse median spaces and their intervals.
Selecta Mathematica, 27 (2), [20].
(doi:10.1007/s00029-021-00623-8).
Abstract
This paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of δ-hyperbolicity.
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THE-INTRINSIC-GEOMETRY-OF-COARSE-MEDIAN-SPACES-AND-THEIR-INTERVALS_Final
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Accepted/In Press date: 21 June 2020
Published date: May 2021
Additional Information:
Funding Information:
J. Zhang was supported for this research by a Fellowship from the Sino-British Trust, International Exchanges 2017 Cost Share (China) Grant EC/NSFC/170341, NSFC11871342 and NSFC11811530291.
Publisher Copyright:
© 2021, The Author(s).
Keywords:
Coarse interval structures and a canonical metric, Coarse median spaces, Median algebras
Identifiers
Local EPrints ID: 442164
URI: http://eprints.soton.ac.uk/id/eprint/442164
ISSN: 1022-1824
PURE UUID: 0029459a-9ff7-4539-b2bb-1a8eb583f47d
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Date deposited: 08 Jul 2020 16:30
Last modified: 17 Mar 2024 05:42
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Author:
Jiawen Zhang
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