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

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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.