Homogeneous length functions on groups

Fritz, Tobias; Gadgil, Siddhartha; Khare, Apoorva; Nielsen, Pace; Silberman, Lior; Tao, Terence

doi:10.2140/ant.2018.12.1773

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Homogeneous length functions on groups

Tobias Fritz, Siddhartha Gadgil, Apoorva Khare, Pace P. Nielsen, Lior Silberman and Terence Tao

Vol. 12 (2018), No. 7, 1773–1786

DOI: 10.2140/ant.2018.12.1773

Abstract

A pseudolength function defined on an arbitrary group $G = (G, \cdot, e, {()}^{- 1})$ is a map $ℓ : G \to [0, + \infty)$ obeying $ℓ (e) = 0$ , the symmetry property $ℓ (x^{- 1}) = ℓ (x)$ , and the triangle inequality $ℓ (x y) \leq ℓ (x) + ℓ (y)$ for all $x, y \in G$ . We consider pseudolength functions which saturate the triangle inequality whenever $x = y$ , or equivalently those that are homogeneous in the sense that $ℓ (x^{n}) = n ℓ (x)$ for all $n \in ℕ$ . We show that this implies that $ℓ ([x, y]) = 0$ for all $x, y \in G$ . 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.

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Keywords

homogeneous length function, pseudolength function, quasimorphism, Banach space embedding

Mathematical Subject Classification 2010

Primary: 20F12

Secondary: 20F65

Milestones

Received: 11 January 2018

Revised: 20 April 2018

Accepted: 12 June 2018

Published: 27 October 2018

Authors

Tobias Fritz
Max Planck Institute for Mathematics in the Sciences Leipzig Germany

Siddhartha Gadgil
Department of Mathematics Indian Institute of Science Bangalore India

Apoorva Khare
Department of Mathematics Indian Institute of Science Bangalore India	Analysis and Probability Research Group Indian Institute of Science Bangalore India

Pace P. Nielsen
Department of Mathematics Brigham Young University Provo, UT United States

Lior Silberman
University of British Columbia Vancouver BC Canada

Terence Tao
Department of Mathematics University of California Los Angeles Los Angeles, CA United States

Vol. 12, No. 7, 2018