Homogeneous length functions on groups
| Autor: | Fritz, T, Gadgil, S, Khare, A, Nielsen, PP, Silberman, LA, Tao, T, Polymath, DHJ |
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| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
pseudolength function
20F12 20F12 20F65 Banach space Group Theory (math.GR) 01 natural sciences homogeneous length function Combinatorics Mathematics - Geometric Topology 010104 statistics & probability Mathematics - Metric Geometry FOS: Mathematics 0101 mathematics 20F65 Mathematics Algebra and Number Theory Triangle inequality 010102 general mathematics Geometric Topology (math.GT) Metric Geometry (math.MG) Functional Analysis (math.FA) Mathematics - Functional Analysis Banach space embedding Homogeneous quasimorphism Mathematics - Group Theory |
| Zdroj: | Algebra Number Theory 12, no. 7 (2018), 1773-1786 |
| Popis: | A pseudo-length function defined on an arbitrary group $G = (G,\cdot,e, (\,)^{-1})$ is a map $\ell: G \to [0,+\infty)$ obeying $\ell(e)=0$, the symmetry property $\ell(x^{-1}) = \ell(x)$, and the triangle inequality $\ell(xy) \leqslant \ell(x) + \ell(y)$ for all $x,y \in G$. We consider pseudo-length functions which saturate the triangle inequality whenever $x=y$, or equivalently those that are homogeneous in the sense that $\ell(x^n) = n\,\ell(x)$ for all $n\in\mathbb{N}$. We show that this implies that $\ell([x,y])=0$ for all $x,y \in G$. This leads to a classification of such pseudo-length 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. Modified Proposition 2.1 (see Remark 2.5), with a "quasified" application in Theorem 4.4. The paper is also streamlined. 14 pages, no figures, to appear in "Algebra & Number Theory" |
| Databáze: | OpenAIRE |
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