Self-similar groups of rooted tree automorphisms

建议的研究计划以自相似群体作用于有根树的特性为中心。用最一般的术语来说，作用于有根、球状同质树的群理论可以理解为通过使用拓扑和几何的语言、方法、思想和直觉来研究剩余有限群。作用对象（树）的自相似性和作用于有根树的群理论中固定顶点（根）的存在导致了一组自然有限性条件，例如由有限自相似集生成，具有有限索引的刚性稳定器，由有限多个禁止树模式定义，具有有限核等。这些条件在涉及自相似群表示问题的提出的问题和研究方向中起着至关重要的作用， Bieri-Neumann-Strebel Sigma 不变量、虚拟自同态及其应用、树上的有限约束群和其他群移位、与 Hausdorff 维数的关系以及算法问题，特别关注共轭问题。在数学内外的许多努力中，理解是通过两个通常相互交织的阶段来实现的。也就是说，在第一阶段，人们寻求对某些类别的对象和情况的理解，这些对象和情况以其简单性或规律性而著称，而在第二阶段，人们寻求理解它们组合在一起以构建或至少近似更复杂的对象和情况的方式。由于就其本质而言，自相似性的概念涉及在实体本身内可以以不同尺度找到原始副本的实体，因此从更简单和更规则的结构构建/理解复杂自相似结构的方法似乎特别适合。拟议的研究有助于理解有根树上的自相似群体行为的两个自然阶段。例如，所有有限自相似群都被表征，并且它们充当有限约束群以及更一般地所有自相似群被组合在一起的构建块。另一方面，算法问题的可判定性是在构建块已经被很好理解的背景下探索的，并且简单实例中问题的解决方案可能被组装成复合结构中的解决方案。

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.