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Homogeneous length functions on groups

Fritz, T and Gadgil, S and Khare, A and Nielsen, PP and Silberman, LA and Tao, T and Polymath, DHJ (2018) Homogeneous length functions on groups. In: Algebra and Number Theory, 12 (7). pp. 1773-1786.

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Official URL: https://doi.org/ 10.2140/ant.2018.12.1773


A pseudolength function defined on an arbitrary group G â� (G, ·, e, ( ) â��1 ) is a map â��: G â�� 0, +â��) obeying â��(e) â� 0, the symmetry property â��(x â��1 ) â� â��(x), and the triangle inequality â��(xy) â�¤ â��(x) + â��(y) for all x, y â�� G. We consider pseudolength functions which saturate the triangle inequality whenever x â� y, or equivalently those that are homogeneous in the sense that â��(x n ) â� nâ��(x) for all n â�� â��. We show that this implies that â��(x, y]) â� 0 for all x, y â�� G. This leads to a classification of such pseudolength functions as pullbacks from embeddings into a Banach space. We also obtain a quantitative version of our main result which allows for defects in the triangle inequality or the homogeneity property. © 2018, Mathematical Sciences Publishers. All rights reserved.

Item Type: Journal Article
Publication: Algebra and Number Theory
Publisher: Mathematical Sciences Publishers
Additional Information: Copyright for this article belongs to Mathematical Sciences Publishers.
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 10 Apr 2019 06:00
Last Modified: 10 Apr 2019 06:00
URI: http://eprints.iisc.ac.in/id/eprint/62128

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