Vol. 12, No. 7, 2018

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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
Abstract

A pseudolength function defined on an arbitrary group G = (G,,e,()1) is a map : G [0,+) obeying (e) = 0, the symmetry property (x1) = (x), and the triangle inequality (xy) (x) + (y) for all x,y G. We consider pseudolength functions which saturate the triangle inequality whenever x = y, or equivalently those that are homogeneous in the sense that (xn) = n(x) for all n . We show that this implies that ([x,y]) = 0 for all x,y 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.

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