Growth of periodic Grigorchuk groups

• 成果类型: Article
• 署名作者: Erschler, Anna; Zheng, Tianyi
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS); University of California System; University of California San Diego
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00922-0
• 发表日期: 2020
• 页码: 1069-1155
• 关键词: random-walks; upper-bounds; jump-processes; markov-chains; constants

摘要

On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the first Grigorchuk group G we show that its growth vG,S(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{G,S}(n)$$\end{document} satisfies limn ->infinity loglogvG,S(n)/logn=alpha 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{n\rightarrow \infty }\log \log v_{G,S}(n)/\log n=\alpha _{0}$$\end{document}, where alpha 0=log2log lambda 0 approximate to 0.7674\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{0}=\frac{\log 2}{\log \lambda _{0}}\approx 0.7674$$\end{document}, lambda 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{0}$$\end{document} is the positive root of the polynomial X3-X2-2X-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X<^>{3}-X<^>{2}-2X-4$$\end{document}.