Self-similar groups of rooted tree automorphisms

建议的研究计划集中在作用于有根树的自相似群的特性上。一般来说，作用于有根的球齐次树的群的理论可以被理解为利用拓扑和几何的语言、方法、思想和直觉对剩余有限群的研究。作用于有根树的群理论中，由于作用对象（树）的自相似性和固定顶点（根）的存在，导致了一组自然有限条件，如由有限自相似集生成、具有有限指标的刚性稳定器、由有限多个禁止树模式定义、具有有限核等。这些条件在自相似群的表示、Bieri-Neumann-Strebel Sigma不变量、虚拟自同态及其应用、有限约束群和树上其他群的移动、与Hausdorff维数的关系以及算法问题中起着至关重要的作用，并特别关注共轭问题。在数学内外的许多努力中，理解通常是在两个相互交织的阶段中实现的。也就是说，在第一阶段，人们寻求理解某些类别的对象和情况，这些对象和情况以其简单性或规律性而区别开来，在第二阶段，人们寻求理解它们组合在一起构建或至少近似更复杂的对象和情况的方式。因为，就其本质而言，自相似的概念涉及实体，其中原始副本可以在实体本身的各种尺度上找到，从更简单和更规则的结构中构建/理解复杂的自相似结构的方法似乎特别适合。提出的研究有助于理解有根树的自相似群体行为的两个自然阶段。例如，所有有限的自相似群都是有特征的，它们是组成有限约束群和所有自相似群的基础。另一方面，在构建块已经被很好地理解并且简单实例中的问题的解可能被组合成复合结构中的解的情况下，探索算法问题的可判定性。

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.