(1) Computation with polycyclic groups
(2) Representations and cohomology for polycyclic groups
(1) This talk will provide a brief overview of polycyclic groups and methods to compute with them. Even though polycyclic groups are often described in terms of group presentations (- a computational model where many problems are known to be computationally undecidable -), polycyclic groups can be described by so-called polycyclic presentations and there exist powerful algorithmic tools to work with such presentations. We will see some interesting examples, including: solutions to the conjugacy problem in polycyclic groups, and how to compute the Frattini subgroup of a polycyclic group.
(2) Continuing with the topic of polycyclic groups, in this talk we will see how representation theory and group cohomology can be used in computations with polycyclic groups. The latter leads to an efficient algorithm to construct certain group extensions of polycyclic groups.
Finite permutation quotients of infinite groups: Algorithmic connection in both directions
With algorithms for groups, finite (permutation and matrix) groups, and finitely presented groups are almost two worlds. Methods used for one are, at first glance, entirely disjoint from those used for the other, with only polycyclic presentations (e.g. for finite solvable groups) relating to both.
I want to show that this is in fact a rather distorted picture: When working with finitely presented groups, homomorphisms into (finite) groups can become a fundamental tool. (This holds even more if they are isomorphisms, as for example arises for classical groups over rings of integers, such as $SL_n(\Z)$.
Vice versa, considering a finite group $G$ as quotient of a free group naturally brings up extensions of the form $M.G$, and algorithms that produce normal forms will give rise to effective methods of computing such extensions and, more generally, certain maximal extensions, called universal covers. These covers in turn are a principal tool in finding certain quotients of finitely presented groups, as in the celebrated p-Quotient algorithm, and the construction of finite groups.
I want to show the ideas (and basic algorithms) behind these connections, and hope to be able to indicate how these methods could extend to other classes of groups. Rather than aiming for description of algorithms to the point of implementation, I want to show in examples how these methods (and their implementations) can be used in practice.
Some of the methods I will show are by now almost classical, some could be considered as obvious generalization from polycyclic groups, and some arose in collaborations with Alla Detinko, Heiko Dietrich and Dane Flanery. What is new(ish) is how to tie it all together.
Computably t.d.l.c. groups
In the first talk I will introduce two notions of computable presentation of a t.d.l.c. group, and show their equivalence. One relies on standard notions of computability in the uncountable setting. The other restricts computation to a countable structure of approximations of the elements, the ``meet groupoid” of compact open cosets. Based on this, I obtain various examples of computably t.d.l.c. groups, such as $Aut(T_d)$ and some algebraic groups over the field of $p$-adic numbers. The first talk also outlines the computability theoretic notions that are needed.
In the second talk I show that given a computable presentation of a t.d.l.c. group, the modular function and the Cayley-Abels graphs (in the compactly generated case) are computable. I discuss the open question whether also the scale function is computable.
I will give a criterion when the computable presentation is unique up to computable isomorphism. I explain why the class of computably t.d.l.c. groups is closed under most of the constructions studied by Wesolek.
Reference:
A. Melnikov and A. Nies. Computably totally disconnected locally compact groups. Preprint, 2022, https://arxiv.org/pdf/2204.09878.pdf
First order rigidity of profinite groups
In recent years the model theory of profinite groups has seen increased attention.
In the first talk I will explain the expressive power of first-order language in the context of profinite groups. A group is called first-order rigid within a given class of groups if it can be described by first-order logic up to isomorphism. I will give examples of this phenomenon for profinite groups and explain its limitations.
The notion of Vapnik-Chervonenkis (VC) dimension of a set of functions is crucial in the context of machine learning. This dimension is finite if and only if the set is 'learnable' and this corresponds exactly to a first-order theory not having the independence property (NIP). In joint work with D. Macpherson we showed that a profinite group G is learnable in this sense if and only if G has an open normal subgroup which is a finite direct product of compact p-adic analytic groups. I will explain the ingredients for this result and sketch the proof.