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Weakly multiplicative arithmetic functions and the normal growth of groups

Authors
  • Schlage-Puchta, Jan-Christoph1
  • 1 Universität Rostock, Rostock, Germany , Rostock (Germany)
Type
Published Article
Journal
Archiv der Mathematik
Publisher
Springer International Publishing
Publication Date
Nov 23, 2018
Volume
112
Issue
3
Pages
233–240
Identifiers
DOI: 10.1007/s00013-018-1267-9
Source
Springer Nature
Keywords
License
Yellow

Abstract

We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or log\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log $$\end{document}-uniformly continuous normal order is close to a function of the form n↦nc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\mapsto n^c$$\end{document}. As an application we show that a finitely generated, residually finite, infinite group, whose normal growth has a non-decreasing or a log\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log $$\end{document}-uniformly continuous normal order, is isomorphic to (Z,+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {Z}, +)$$\end{document}.

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