On Loops Intersecting at Most Once
On Loops Intersecting at Most Once
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关于最多相交一次的循环
DOI：
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发表时间：
2019
影响因子：
2.2
通讯作者：
J. Greene
中科院分区：
文献类型：
--
作者：
J. Greene
We prove that on a closed, orientable surface of genus g, the maximum cardinality of a set of simple loops with the property that no two are homotopic or intersect in more than k points grows as a function of g like gk+1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$g^{k+1}$$end{document}, up to a factor of loggdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$log g$$end{document}. The proof of the upper bound uses arguments from probabilistic combinatorics and a theorem of Scott related to the fact that surface groups are LERF.
We prove that on a closed, orientable surface of genus g, the maximum cardinality of a set of simple loops with the property that no two are homotopic or intersect in more than k points grows as a function of g like gk+1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$g^{k+1}$$end{document}, up to a factor of loggdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$log g$$end{document}. The proof of the upper bound uses arguments from probabilistic combinatorics and a theorem of Scott related to the fact that surface groups are LERF.