Finiteness properties of groups acting on rooted trees

提议者提出了一个研究计划，涉及作用于有根树的自相似群的性质，特别注意享受额外有限性性质的自相似群类，例如由有限自相似集生成的分支群（具有有限指数的刚性稳定器），受约束的（被许多禁止的模式定义），收缩（有有限的核），被扭转，等等。在建议的研究方向是涉及增长问题的问题，符号动力学中的概念和结果与自相似群理论之间的对应关系，自相似群闭包的Hausdorff维数，与自相似群设置中的扭转、增长、Hausdorff维数和分支属性有关的问题，以及一些特别有趣的群的性质，如河内Towers群和tent map群。广义地说，自相似性的概念涉及一个实体，在这个实体中，可以在实体本身的各种尺度上找到原始实体的许多副本。这是一个基本的概念，反映在许多方面，无论是在自然界（众所周知的例子包括云的结构，海岸形状，植物分枝模式等）。在数学中（如递归、迭代、自引用、重正化等）。自相似性的概念最近被引入到群论中，通过对有根树的作用。通过对有根树的作用来研究群的某些方面和某些类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，同时简化了符号和表示。事实上，正是这种语言的转变，使更好的直觉，并导致持续爆炸的见解，突破，并与其他数学领域的联系。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，例如中国环，现代数学结构，例如群中间增长，以及全新的概念，例如迭代单值群）。

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).