Tag - Generation questions for groups

Umberto Zannier: Bounded generation in linear groups and exponential parametrizations

31st December, 2022

In fairly recent joint work with Corvaja, Rapinchuk, Ren, we applied results from Diophantine S-unit theory to problems of 'bounded generation' in linear groups: this property is a strong form of finite generation and is useful for several issues in the setting. Focusing on 'anisotropic groups' (i.e. containing only semi-simple elements), we could give a simple essentially complete description of those with the property. More recently, in further joint work also with Demeio, we proved the natural expectation that sets boundedly generated by semi-simple elements (in linear groups over number fields) are 'sparse'. Actually, this holds for all sets obtained by exponential parametrizations. As a special consequence, this gives back the previous results with a different approach and additional precision and generality.

Scott Harper: Invariable generation and totally deranged elements of simple groups

29th July, 2022

By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.

Andrei Rapinchuk: Groups with bounded generation: old and new

6th May, 2021

A group is said to have bounded generation (BG) if it is a finite product of cyclic subgroups. We will survey the known examples of groups with (BG) and their properties. We will then report on a recent result (joint with P. Corvaja, J. Ren and U. Zannier) that non-virtually abelian anisotropic linear groups (i. e. those consisting entirely of semi-simple elements) are not boundedly generated. The proofs rely on number-theoretic techniques.

Charles Cox: Spread and infinite groups

25th January, 2021

My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are various natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, and Scott Harper and Casey Donoven have raised several intriguing questions for spread for infinite groups. A group G has spread k if for every g1,…,gk we can find an h in G such that ⟨gi,h⟩=G. For any group we can say that if it has a proper quotient that is non-cyclic, then it has spread 0. In the finite world there is then the astounding result - which is the work of many authors - that this condition on proper quotients is not just a necessary condition for positive spread, but is also a sufficient one. Harper-Donoven's first question is therefore: is this the case for infinite groups? Well, no. But that’s for the trivial reason that we have infinite simple groups that are not 2-generated (and they point out that 3-generated examples are also known). But if we restrict ourselves to 2-generated groups, what happens? In this talk we'll see the answer to this question. The arguments will be concrete and accessible to a general audience.

Daniele Garzoni: Invariable generation of finite simple groups

25th October, 2020

A subset X of a group G invariably generates G if we are free to replace each element of X by an arbitrary conjugate, and we must always obtain a generating set of G. This concept was introduced by Dixon in the early nineties with motivations from computational Galois theory. We will review these motivations and their intimate connections with permutation groups. We will then present some new results concerning the probability of generating invariably a finite simple group. For instance, we will see that two random elements of a finite simple group of Lie type of bounded rank invariably generate with probability bounded away from zero.