Growth and homogeneity of matchbox manifolds

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• 作者：Jessica Dyer^a ; ^{jesscdyer@gmail.com} ; Steven Hurder^b ; ^{hurder@uic.edu} ; Olga Lukina^b ; ^{lukina@uic.edu}
• 关键词：Non-abelian group actions on Cantor sets ; Growth of leaves ; Group chains ; Regularity ; Homogeneity
• 刊名：Indagationes Mathematicae
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：28
• 期：1
• 页码：145-169
• 全文大小：363 K
• 卷排序：28

文摘

A matchbox manifold with one-dimensional leaves which has equicontinuous holonomy dynamics must be a homogeneous space, and so must be homeomorphic to a classical Vietoris solenoid. In this work, we consider the problem, what can be said about a matchbox manifold with equicontinuous holonomy dynamics, and all of whose leaves have at most polynomial growth type? We show that such a space must have a finite covering for which the global holonomy group of its foliation is nilpotent. As a consequence, we show that if the growth type of the leaves is polynomial of degree at most 3, then there exists a finite covering which is homogeneous. If the growth type of the leaves is polynomial of degree at least 4, then there are additional obstructions to homogeneity, which arise from the structure of nilpotent groups.