Tag - Group theory

Alexandr Zubkov: Harish-Chandra pairs and group superschemes

12th March, 2021

The purpose of my talk is to discuss the following results recently obtained in collaboration with A.Masuoka (Tsukuba University, Japan). First, we prove that a certain category of Harish-Chandra pairs is equivalent to the category of (not necessary affine) locally algebraic group superschemes. Using this fundamental equivalence we superize the famous Barsotti-Chevalley theorem and prove that the sheaf quotient of an algebraic group superscheme over its group super-subscheme is again a superscheme of finite type. I will also formulate some open problems whose solving would bring significant progress in the supergroup theory.

Giles Gardam: Kaplansky’s conjectures

11th March, 2021

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and finish with my recent counterexample to the unit conjecture.

Dawid Kielak: Recognizing surface groups

8th March, 2021

I will address two problems about recognizing surface groups. The first one is the classical problem of classifying Poincaré duality groups in dimension 2. I will present a new approach to this, joint with Peter Kropholler. The second problem is about recognizing surface groups among one-relator groups. Here I will present a new partial result, joint with Giles Gardam and Alan Logan.

Charlotte Hoffmann: Short words of high imprimitivity rank yield hyperbolic one-relator groups

8th March, 2021

It is a long-standing question whether a group of type F that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type F and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than 2, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most 17. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I'll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams.

Tamar Bar-On: Profinite completion of free profinite groups

4th March, 2021

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.

Giles Gardam: Kaplansky’s conjectures

22nd February, 2021

Kaplansky made various related conjectures about group rings, especially for torsion-free groups. For example, the zero divisors conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other group properties such as orderability, and present some recent progress.

Paul-Henry Leemann: Cayley graphs with few automorphisms

22nd February, 2021

Let G be a group and S a generating set. Then the group G naturally acts on the Cayley graph Cay(G,S) by left multiplications. The group G is said to be rigid if there exists an S such that the only automorphisms of Cay(G,S) are the ones coming from the action of G. While the classification of finite rigid groups was achieved in 1981, few results were known about infinite groups. In a recent work, with M. de la Salle we gave a complete classification of infinite finitely generated rigid groups. As a consequence, we also obtain that every finitely generated group admits a Cayley graph with countable automorphism group.

Vladimir Shpilrain: What, if anything, can be done in sublinear time?

4th February, 2021

In his talk on September 10, 2020, Yuri Gurevich discussed some algorithms that run in linear time (in the "length" of an input). We are going to take it up a notch and discuss what can be done in sublinear time; in particular, without reading the whole input but only a small part thereof. One well-known example is deciding divisibility of a decimal integer by 2, 5, or 10: this is done by reading just the last digit. We will discuss some less obvious examples from (semi)group theory.