Tag - Infinite groups

Michael Wibmer: Expansive endomorphisms of profinite groups

25th May, 2023

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.

Emma Brink: Condensed Group Cohomology

18th April, 2023

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.

Martina Conte: Definability of the rank and the dimension of p-adic analytic pro-p groups

31st March, 2023

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.

Agatha Atkarskaya: Introduction to group-like small cancellation theory for rings

18th November, 2022

The structure of small cancellation groups is well known. Тhey are widely used in construction of groups with unusual properties (for example Burnside groups and Tarskii monster). We were interested in developing a similar theory for rings. However, such theory meets significant difficulties because, unlike groups, rings have two operations: addition and multiplication. I will speak about small cancellation conditions for rings that we introduced. These conditions provide the desired properties. I will discuss our way towards these conditions, examples and possible applications of small cancellation rings.

Laurent Bartholdi: Dimension series and homotopy groups of spheres

28th October, 2022

The lower central series of a group G is defined by γ1=G and γn = [G,γn-1]. The 'dimension series', introduced by Magnus, is defined using the group algebra over the integers:

δn = { g : g-1 belongs to the nth power of the augmentation ideal }.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has δn ≥ γn, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with δ4 / γ4 cyclic of order 2. On the positive side, Sjogren showed that δn / γn is always a torsion group, of exponent bounded by a function of n. Furthermore, it was believed (and falsely proved by Gupta) that only 2-torsion may occur.

In joint work with Roman Mikhailov, we prove however that every torsion abelian group may occur as a quotient δn / γn; this proves that Sjogren's result is essentially optimal.

Even more interestingly, we show that this problem is intimately connected to the homotopy groups πn(Sm) of spheres; more precisely, the quotient δn / γn is related to the difference between homotopy and homology. We may explicitly produce p-torsion elements starting from the order-p element in the homotopy group π2p(S2) due to Serre.

Chen Meiri: Word Width in Higher Rank Arithmetic Groups

18th October, 2022

A word on d letters is an element of the free group of rank d, say, with basis x1,…,xd. Given a word w=w(x1,…,xd) on d letters, for every group G, there is a word map w:Gd→ G given by substituting the xi with elements of G. We say that a word w has a finite width n in the group G if any element in the subgroup generated by w(G) is a product of at most n element of w(G) or their inverses. In this talk, I will survey results about word width in several families of groups and then restrict the focus to the family of higher rank arithmetic groups. I will present a conjecture about word width in higher-rank arithmetic groups and explain some consequences, most notably, to the Congruence Subgroup Problem.

Nikolay Nikolov: On conjugacy classes of profinite groups

10th October, 2022

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.

Eric Freden: Aspects of growth in Baumslag-Solitar groups

7th October, 2022

In 1997, Grigorchuk and de la Harpe suggested computing the growth series for the Baumslag-Solitar group BS(2,3). After 25 years, this is still an open problem. In fact, the growth of only the solvable groups BS(1,n) and automatic groups BS(n,n) are known. In this talk I will review what has since been discovered about these remarkable groups and conclude with new unpublished results concerning the exponents of growth for the subfamily BS(2,2n).

Kane Townsend: Hyperbolic groups with k-geodetic Cayley graphs

7th October, 2022

A locally-finite simple connected graph is said to be k-geodetic for some k ≥ 1, if there are at most k distinct geodesics between any two vertices of the graph. We investigate the properties of hyperbolic groups with k-geodetic Cayley graphs. To begin, we show that k-geodetic graphs cannot have a "ladder-like" geodesic structure with unbounded length. Using this bound, we generalize a well-known result of Papasoglu that states hyperbolic groups with 1-geodetic Cayley graphs are virtually free. We then investigate which elements of the hyperbolic group with k-geodetic Cayley graph commute with a given infinite order element.