Finiteness properties of groups acting on rooted trees

作者提出了一个涉及作用在根树上的自相似群的性质的研究计划，特别注意具有附加有限性的一类自相似群，如由有限自相似集生成的，是分枝群(具有有限指数的刚性稳定器)，被有限约束(由有限多个禁止模式定义)，被压缩(具有有限核)，被扭转等.在所提出的研究方向中，涉及增长问题，符号动力学和自相似群理论中的概念和结果之间的对应，自相似群的闭包的Hausdorff维，与扭转，增长，Hausdorff维数和自相似群环境下的分枝性质，以及一些特别有趣的群的性质，如Hanoi Towers群和帐篷映射群。广义地说，自相似的概念涉及一个实体，在该实体中可以在不同的尺度上找到原始的许多副本。这是一个在许多方面反映在自然界中的基本概念(众所周知的例子包括云的结构、海岸形状、植物分枝图案等)。在数学上(作为递归、迭代、自引用、重整化等)。最近，通过对有根树的作用，自相似的概念被引入群论。通过在有根的树上的动作来研究某些方面和某些类别的组的观点已被证明是相当有成效的，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，正是语言的这种转变使人们有了更好的直觉，并导致了洞察力、突破性成果和与其他数学领域的链接的持续爆炸。目前的研究方向是多方面的，反映了这一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来解释和联系一些2000年前的问题，如中国环，现代数学结构，如中等增长的群，以及全新的概念，如迭代单元群)。

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).