In this talk, I will report on joint work with Barnes-Barthel and Barthel-Greenlees which analyses the category of rational G-equivariant spectra for G a profinite group or compact Lie group respectively. In particular, I will focus on a series of results regarding the Balmer spectra of these categories, and how the topology of these topological spaces informs structural results regarding the category.
Étale algebraic groups over a field k are equivalent to finite groups with a continuous action of the absolute Galois group of k. The difference version of this well-known result asserts that étale difference algebraic groups over a difference field k (i.e., a field equipped with an endomorphism) are equivalent to profinite groups equipped with an expansive endomorphism and a certain compatible difference Galois action. In any case, understanding the structure of expansive endomorphisms of profinite groups seems a worthwhile endeavour and that's what this talk is about.
The theory of condensed sets, developed by Dustin Clausen and Peter Scholze, provides a framework well-suited to study algebraic objects that carry a topology. In my talk, I will discuss the basic properties of the cohomology of condensed groups and its relation to continuous group cohomology. Johannes Anschütz and Arthur-César le Bras showed that for locally profinite groups and solid (e.g. discrete) coefficients, condensed group cohomology agrees with continuous group cohomology. On the other hand, if G is a locally compact and locally contractible topological group (e.g., a Lie group), and M is a discrete group with trivial G-action, then the condensed group cohomology of G, M (the sheaves of continuous functions into G and M) is isomorphic to the singular cohomology of the classifying space of G with coefficients in M, whereas the continuous group cohomology of G with coefficients in M is isomorphic to the singular cohomology of the classifying space of π0(G) with coefficients in M.
Generalizing results of Johannes Anschütz and Arthur-César le Bras on locally profinite groups, I will explain that continuous group cohomology with solid coefficients can be described as a cohomological δ-functor in the condensed setting for a large class of topological groups.
Recently, Nies, Segal and Tent started an investigation of finite axiomatizability in the realm of profinite groups. Among the classes of profinite groups under their consideration is the class of p-adic analytic pro-p groups. In joint work with Benjamin Klopsch, we consider two key invariants of these groups, namely rank and dimension, and show that they can be characterized by a single first-order sentence. Before discussing these results I will introduce the relevant background. If time permits, I will also present some natural generalisations.
Free products with amalgamation and HNN-extensions are two main constructions of combinatorial group theory. I shall discuss these two constructions in the category of pro-p groups, presenting results on splittings of pro-p groups as an amalgamated free pro-p product or a pro-p HNN-extension and relating them with pro-p version of Bass-Serre's theory of groups acting on trees. I shall also compare the pro-p results with similar results for abstract groups.
It is well-known that the number of conjugacy classes of a finite group G tends to infinity as the size of G tends to infinity. There is no such result for a general infinite group. In this talk I will discuss the situation when G is a profinite group and show that the number of conjugacy of G is then uncountable unless G is finite. The proof depends on many classical results on finite groups and in particular the classification of the finite simple groups.
A profinite group is called strongly complete if every subgroup of finite index is open. Strongly complete groups are very useful, since in such groups the algebra determines the topology. For example, every homomorphism from a strongly complete group to any profinite group is continuous, and thus a homomorphism in the category of profinite groups. For many years it was an open question, whether every finitely generated profinite group is strongly complete. In 2000 Segal and Nikolov published a positive proof for this conjecture. In the talk we present the general idea of the proof, and show some nice results relying on this theorem.
The profinite completion of a free profinite group on infinite set of generators is a profinite group of greater rank. However, it is still unknown whether it is a free profinite group too. I am going to present some partial results regarding to this question, which is equivalent to ask: what abstract embedding problems can a free profinite group solve.
You must be logged in to post a comment.