Asymptotic dimension and small subsets in locally compact topological groups

被引：0
作者：
Taras Banakh
Ostap Chervak
Nadya Lyaskovska
机构：
[1] Jan Kochanowski University,
[2] Ivan Franko National University of Lviv,undefined
来源：
Geometriae Dedicata | 2014年 / 169卷
关键词：
Asymptotic dimension; Locally compact group; Coarse structure; Small set; 20F65; 20F69; 54F45; 55M10;
D O I：
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学科分类号：
摘要：
We prove that for a coarse space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} the ideal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S }(X)$$\end{document} of small subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} coincides with the ideal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{D }_<(X)=\{A\subset X:\mathrm{asdim}(A)<\mathrm{asdim}(X)\}$$\end{document} provided that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} is coarsely equivalent to a Euclidean space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R }^n$$\end{document}. Also we prove that for a locally compact Abelian group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document}, the equality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S }(X)=\mathcal{D }_<(X)$$\end{document} holds if and only if the group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} is compactly generated.
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页码：383 / 396
页数：13
相关论文
共 13 条
[1]  
Banakh T(2010)The coarse classification of countable abelian groups Trans. Am. Math. Soc. 362 4755-4780
[2]  
Higes J(2010)Completeness of translation-invariant ideals in groups Ukr. Mat. Zh. 62 1022-1031
[3]  
Zarichnyi I(1960)Wreath products and finitely presented groups Math. Z. 75 22-28
[4]  
Banakh T(2001)On certain subsets of a group II Q &A Gen. Topol. 19 81-94
[5]  
Lyaskovska N(2007)A remark on asymptotic dimension and digital dimension of finite metric spaces Mat. Stud. 27 100-104
[6]  
Baumslag G.(2006)Asymptotic dimension of discrete groups Fund. Math. 189 27-34
[7]  
Bella A(2008)Asymptotic dimension of finitely presented groups Proc. Am. Math. Soc. 136 4103-4110
[8]  
Malykhin VI(undefined)undefined undefined undefined undefined-undefined
[9]  
Chatyrko V(undefined)undefined undefined undefined undefined-undefined
[10]  
Zarichnyi M(undefined)undefined undefined undefined undefined-undefined
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