火柴盒流形的增长性和齐性
Growth and homogeneity of matchbox manifolds
摘要：
具有等度连续完整动力学的一维叶火柴盒流形必须是一个齐性空间，因此必须同胚于经典的Vietoris螺线管。在这项工作中，我们考虑的问题，什么可以说是一个火柴盒流形等度连续holonomy动态，其所有的叶子有最多多项式增长型？我们证明了这样的空间必须有一个有限覆盖，使得它的叶理的整体和乐群是幂零的。因此，我们证明了，如果叶的生长类型是多项式的次数至多为3，那么存在一个有限覆盖是齐次的。如果叶的生长类型是至少4次的多项式，则存在对齐性的额外障碍，其产生于幂零群的结构。
英文摘要：
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.
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