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Homogeneous length functions on groups

Fritz, Tobias and Gadgil, Siddhartha and Khare, Apoorva and Nielsen, Pace P and Silberman, Lion and Tao, Terence (2018) Homogeneous length functions on groups. In: ALGEBRA & NUMBER THEORY, 12 (7). pp. 1773-1786.

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Official URL: http://dx.doi.org/10.2140/ant.2018.12.1773


A pseudolength function defined on an arbitrary group G = (G, . , e, ()(-1)) is a map l : G -> 0, +infinity) obeying l(e) = 0, the symmetry property l (x(-1)) = l(x), and the triangle inequality l(xy) <= l(x) + l(y) for all x, y is an element of G. We consider pseudolength functions which saturate the triangle inequality whenever x = y, or equivalently those that are homogeneous in the sense that l(x(n)) = nl(x) for all n is an element of N. We show that this implies that l(x, y]) = 0 for all x, y is an element of 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.

Item Type: Journal Article
Additional Information: Copy right for this article belong to MATHEMATICAL SCIENCE PUBL
Keywords: homogeneous length function; pseudolength function; quasimorphism; Banach space embedding
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 30 Nov 2018 14:49
Last Modified: 30 Nov 2018 14:49
URI: http://eprints.iisc.ac.in/id/eprint/61200

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