Self-similar groups of rooted tree automorphisms
有根树自同构的自相似群
基本信息
- 批准号:1105520
- 负责人:
- 金额:$ 12.28万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2011
- 资助国家:美国
- 起止时间:2011-09-01 至 2014-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The suggested research plan centers around properties of self-similar groups acting on rooted trees. In most general terms, the theory of groups acting on rooted, spherically homogeneous, trees can be understood as study of residually finite groups by using the language, methods, ideas and the intuition from topology and geometry. The self-similarity of the object of action (the tree) and the presence of a fixed vertex (the root) in the theory of groups acting on rooted trees lead to a set of natural finiteness conditions, such as being a generated by a finite self-similar set, having rigid stabilizers of finite index, being defined by finitely many forbidden tree patterns, having finite nucleus, etc. Such conditions play a crucial role in the proposed problems and directions of study involving questions on presentations of self-similar groups, Bieri-Neumann-Strebel Sigma invariants, virtual endomorphisms and their applications, finitely constrained groups and other group shifts on trees, relations to Hausdorff dimension, and algorithmic problems, with special attention given to the conjugacy problem. In many endeavors, in and outside of mathematics, understanding is achieved in two, often intertwined, phases. Namely, in the first phase one seeks understanding of some classes of objects and situations distinguished by their simplicity or regularity, and in the second understanding of the ways in which they fit together to build, or at least approximate, the more complex ones. Since, by its very nature, the notion of self-similarity concerns entities in which copies of the original can be found at various scales within the entity itself, the approach of building/understanding complex self-similar structures from simpler and more regular ones seems particularly well suited. The proposed research contributes to both natural phases in the understanding of self-similar group actions on rooted trees. For instance, all finite self-similar groups are being characterized, and they serve as the building blocks from which the finitely constrained groups, and more generally all self-similar groups, are put together. On the other hand, the decidability of algorithmic questions is explored in contexts in which the building blocks are already well understood and the solution of the problem in the simple instances could possibly be assembled into a solution in the composite structure.
建议的研究计划围绕自相似群体作用于有根树的特性展开。用最一般的话来说,作用在有根、球齐次树上的群理论可以理解为使用拓扑学和几何学的语言、方法、思想和直觉来研究剩余有限群。作用对象的自相似性(树)和固定顶点的存在作用于有根树的群理论中的根导致了一组自然有限性条件,例如由有限自相似集生成,具有有限指数的刚性稳定子,由许多禁止树模式定义,具有有限核,这些条件在所提出的问题和研究方向中起着至关重要的作用,这些问题涉及自相似群的表示,Bieri-Neumann-Strebel Sigma不变量,虚自同态及其应用,树上的约束群和其他群移位,与Hausdorff维数的关系,以及算法问题,特别注意共轭问题。在数学内外的许多努力中,理解是通过两个往往相互交织的阶段来实现的。也就是说,在第一阶段,人们寻求理解某些类别的对象和情况,这些对象和情况以其简单性或规律性而区分,在第二阶段,人们寻求理解它们如何组合在一起以构建或至少近似构建更复杂的对象和情况。由于自相似性的概念本质上涉及实体,其中可以在实体本身的各种尺度上找到原始的副本,因此从更简单和更规则的结构构建/理解复杂的自相似结构的方法似乎特别适合。建议的研究有助于两个自然阶段的理解自相似的群体行动的根树。例如,所有有限的自相似群都被表征,它们作为构建块,从这些构建块中,将有限约束群,更一般地,将所有自相似群放在一起。另一方面,算法问题的可判定性的背景下,其中的积木已经很好地理解和简单的情况下的问题的解决方案可能会组装成一个解决方案的复合结构进行了探讨。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Zoran Sunik其他文献
Topological and Asymptotic Aspects of Group Theory
群论的拓扑和渐近方面
- DOI:
10.1090/conm/394 - 发表时间:
2006 - 期刊:
- 影响因子:0
- 作者:
R. Grigorchuk;M. Mihalik;M. Sapir;Zoran Sunik - 通讯作者:
Zoran Sunik
Branch groups
分支组
- DOI:
10.1007/bf02675625 - 发表时间:
2005 - 期刊:
- 影响因子:0.6
- 作者:
L. Bartholdi;R. Grigorchuk;Zoran Sunik - 通讯作者:
Zoran Sunik
Zoran Sunik的其他文献
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{{ truncateString('Zoran Sunik', 18)}}的其他基金
Conference on Geometric and Probabilistic Methods in Group Theory and Dynamical Systems; College Station, Texas, - November 9-12, 2015
群论和动力系统中的几何和概率方法会议;
- 批准号:
1555792 - 财政年份:2015
- 资助金额:
$ 12.28万 - 项目类别:
Standard Grant
Finiteness properties of groups acting on rooted trees
作用于有根树的群的有限性
- 批准号:
0805932 - 财政年份:2008
- 资助金额:
$ 12.28万 - 项目类别:
Standard Grant
Geometric and probabilistic methods in group theory and dynamical systems
群论和动力系统中的几何和概率方法
- 批准号:
0505808 - 财政年份:2005
- 资助金额:
$ 12.28万 - 项目类别:
Standard Grant
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