Finiteness properties of groups acting on rooted trees

作用于有根树的群的有限性

• 批准号: 0805932
• 负责人: Zoran Sunik
• 金额: $ 10.43万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805932&HistoricalAwards=false
• 关键词: Finiteness; properties; groups; acting; rooted

The proposer suggests a research plan involving properties of self-similar groups acting on rooted trees, with particular attention paid to classes of self-similar groups enjoying additional finiteness properties such as being generated by a finite self-similar set, being branch groups (having rigid stabilizers of finite index), being finitely constrained (being defined by finitely many forbidden patterns), being contracting (having finite nucleus), being torsion, etc. Among the proposed directions of study are problems involving growth questions, correspondence between notions and results in symbolic dynamics and the theory of self-similar groups, Hausdorff dimension of closures of self-similar groups, questions relating torsion, growth, Hausdorff dimension and the branching property in the setting of self-similar groups, and properties of some particularly interesting groups such as Hanoi Towers groups and the tent map groups.Broadly speaking, the notion of self-similarity concerns an entity in which many copies of the original can be found at various scales within the entity itself. This is a fundamental notion reflected in many ways both in nature (well known examples include structure of clouds, coastal shapes, plant branching patterns, etc.) and in mathematics (as a recursion, iteration, self-reference, renormalization, etc.). The notion of self-similarity was recently introduced into group theory through actions on rooted trees. The point of view of studying some aspects and some classes of groups through actions on rooted trees has proved to be rather fruitful, since it allows the introduction of many natural(visual) concepts and ideas, while simplifying the notation and presentation. In fact, it is precisely this shift in language that enabled better intuition and resulted in the ongoing explosion of insights, breakthroughs, and links to other areas of mathematics. The current research moves in many directions, reflecting the richness of the subject and its wide appeal and applicability (we can explain and relate in the language of self-similar groups some 2000 years old problems such as Chinese Rings, modern mathematical constructions such as groups of intermediate growth, and entirely new concepts such as iterated monodromy groups).

提议者提出了一项研究计划，涉及作用于有根树的自相似群的性质，特别关注具有附加有限性质的自相似群的类别，例如由有限自相似集生成、是分支群（具有有限指数的刚性稳定器）、是有限约束的（由有限多个禁止模式定义）、是收缩的（具有有限核）、是扭转的等等。 拟议的研究方向是涉及增长问题、符号动力学和自相似群理论中概念和结果之间的对应关系、自相似群闭包的豪斯多夫维数、有关自相似群设置中的扭转、增长、豪斯多夫维数和分支性质的问题，以及一些特别有趣的群的性质，例如河内塔群和帐篷地图群。 自相似性涉及一个实体，在该实体中可以在实体本身的不同尺度上找到原始的许多副本。这是一个在自然界（众所周知的例子包括云结构、海岸形状、植物分支模式等）和数学（如递归、迭代、自引用、重整化等）中以多种方式反映的基本概念。自相似的概念最近通过对有根树的作用被引入群论。通过对有根树的行动来研究群体的某些方面和某些类别的观点已被证明是相当富有成效的，因为它允许引入许多自然（视觉）概念和想法，同时简化符号和表示。事实上，正是这种语言的转变带来了更好的直觉，并导致见解、突破以及与数学其他领域的联系不断爆炸。当前的研究向多个方向发展，反映了该学科的丰富性及其广泛的吸引力和适用性（我们可以用自相似群的语言解释和联系一些2000年前的问题，如中国环，现代数学结构，如中间增长群，以及全新的概念，如迭代单数群）。