Finiteness properties of groups acting on rooted trees

提出了一个研究计划，涉及作用于有根树上的自相似群的性质，特别关注那些具有额外有限性质的自相似群，如由有限自相似集生成，是分支群（具有有限指标的刚性稳定器），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。提出的研究方向包括涉及生长问题的问题、符号动力学和自相似群理论中概念和结果的对应关系问题、自相似群闭包的Hausdorff维数问题、自相似群设置中有关扭转、生长、Hausdorff维数和分支性质的问题，以及一些特别有趣的群的性质，如Hanoi Towers群和tent map群。广义地说，自相似的概念涉及一个实体，在这个实体内部，可以在不同的尺度上找到许多原件的副本。这是一个基本概念，在自然界（众所周知的例子包括云的结构，海岸形状，植物的分支模式等）和数学（作为递归，迭代，自我参考，重整化等）中都有体现。自相似的概念最近通过对有根树的行为被引入群论。通过对有根的树的动作来研究某些方面和某些类的群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的（视觉的）概念和思想，同时简化了符号和表示。事实上，正是这种语言上的转变使人们有了更好的直觉，并导致了不断涌现的见解、突破，以及与数学其他领域的联系。目前的研究在许多方向上发展，反映了主题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，如迭代一元群）。

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).