A combinatorial approach to the asymptotic theory of pro-p groups via actions on p-adic trees

渐近代数研究的是代数结构的相关参数随结构复杂性的增加而变化的行为。在这里，复杂性可以定义为结构的大小或用于研究给定结构的精度。群的概念是最基本的代数结构之一。群出现在数学及其应用的许多分支中，通常描述给定系统的对称性。经典的例子是多项式方程的解的集合或晶体结构的分类。研究群体的一种常用方法是研究它们的子结构。子群增长理论根据给定指标的子群数目的渐近增长对群进行分类。这一理论自80年代初开始作为纯群论的一个分支而迅速发展。后来，这一理论显示了与数学熵、算术几何和随机过程的关系，并被证明适用于数学的其他部分。计算给定群的子群增长的一种方法是将问题转化为组合问题，并将枚举组合的方法应用于这个新问题。这种方法在处理大群体时相当成功。然而，组合学本质上是一种离散理论，因此这种方法不能直接适用于拓扑群，即不仅组成元素，而且具有距离和收敛性的非平凡概念的代数结构。本项目的目标是使用p进树在这些技术的组合方面引入拓扑结构。这将使我们能够利用过去30年离散群体获得的丰富结果来解决pro-p设置中的问题。当然，一个单独的项目不能翻译整个理论，因此我们关注的是p进树的结构从代数的角度明显可见的情况。一类重要的群是分支群，即在p进树上以非常可传递的方式起作用的群。另一种情况是群对其子群格的作用。这个格不是树，但是，对于大的群，它足够类似于树，所以作用的参数可以通过作用在一个合适的构造树上来近似。虽然该项目是纯群论的一部分，但我们希望与数学的其他分支建立联系。Nottingham群和Demushkin群直接导致伽罗瓦理论的应用，而desins d'enfants和正方形平铺表面应该导致算术几何的应用。

Asymptotic algebra studies the behaviour of relevant parameter of algebraic structures as the complexity of the structure increases. Here, complexity can for example be defined as the size of the structure or the precision used to study a given structure. One of the most basic algebraic structures is the notion of a group. Groups occur in many branches of mathematics and its applications, often describing symmetries of a given system. Classical examples are the set of solutions of a polynomial equation or the classification of crystal structures. One common approach to study groups is by studying their substructures. The theory of subgroup growth classifies groups depending on the asymptotic growth of the number of subgroups of given index. This theory shows rapid development since the early 80s, first as a branch of pure group theory. Later this theory showed relations to sofic entropy, arithmetic geometry, and random processes, and proved to be applicable to other parts of mathematics.One method to compute the subgroup growth of a given group is to translate the problem into a combinatorial problem, and apply methods of enumerative combinatorics to this new problem. This method has been quite successful when dealing with large groups. However, combinatorics is by nature a discrete theory, therefore this approach is not directly applicable to topological groups, that is, algebraic structures in which elements cannot only be composed, but also have non-trivial notions of distance and convergence. The goal of the present project is to use p-adic trees to introduce a topological structure on the combinatorial side of these techniques. This would allow us to exploit the wealth of results obtained for discrete groups in the last 30 years to solve problems in the pro-p setting.Of course, a single project cannot translate a whole theory, therefore we focus on cases where the structure of a p-adic tree is obviously visible from the algebraic point of view. One important class of groups are branch groups, that is, groups, that act in a very transitive way on the p-adic tree. Another case is the action of a group on its subgroup lattice. This lattice is not a tree, however, for large groups it is sufficiently similar to a tree, so the parameters of the action can be approximated by the action on a suitable constructed tree.Although the project is part of pure group theory, we expect relations to other branches of mathematics. Nottingham groups and Demushkin groups directly lead to applications in Galois theory, and dessins d'enfants and square tiled surfaces should lead to applications to arithmetic geometry.