Algebra Seminar

Title: Deficiency, relation gap and 2-dimensional groups.

Abstract: We conjecture a characterisation of the finitely presented residually finite groups with 2-dimensional classifying space in terms of the growth of the deficiency of their finite index subgroups. This conjecture is motivated by an open question about the deficiency gradient of groups and their L²-Betti numbers. In this talk I will relate this conjecture to the relation gap problem for group presentations and establish its higher dimensional analogs.

Joint work with Aditi Kar.

Title: A new generalisation of Camina pairs.

Abstract: Let G be a finite group and N a nontrivial, proper, normal subgroup of G. In 1978 Alan Camina considered pairs (G, N) satisfying the following property: each coset xN ≠ N is contained in a single conjugacy class (necessarily that of x). His aim was to characterise Frobenius groups. The paper spawned much research and the pairs became known as Camina pairs. We relax this condition and just insist each coset contains elements of the same order. We show that in many ways these new pairs resemble Camina pairs, but with some important differences. This is joint work with Alan Camina, Mark L. Lewis, Emanuele Pacifici, Lucia Sanus, and Marco Vergani.

Title: Non-singular identities for finite groups

Abstract: A law (respectively an identity with constants) for a group G is an equation in one or more variables (respectively variables and constants from G) which is satisfied by all tuples of elements from G. Every finite group G satisfies some law, and the length of the shortest law, or the shortest identity with constants, is a natural invariant of G. We survey some of what is known about the asymptotic behaviour of these lengths in various families of finite groups. Motivated by a conjecture of Larsen and Shalev concerning the class of profinite groups satisfying laws, we draw particular attention to the class of "non-singular" identities with constants.

Joint work with Kivanç Ersoy, Jakob Schneider and Andreas Thom.

Title: Twisted conjugacy in dihedral Artin groups

Abstract: Dehn’s famous decision problems for finitely presented groups have been studied for over a century by combinatorial and geometric group theorists. In recent years, a variant of one of these classical problems, namely the twisted conjugacy problem, has been studied. The motivation for this problem comes from Bogopolski, Martino and Ventura who, in 2009, proved an equivalence between conjugacy in group extensions and twisted conjugacy.

In this talk I will give a brief survey of this lesser-known decision problem, and discuss some of the latest results in this area. This includes a framework which can be applied to dihedral Artin groups.

Title: On the Hasse-Norm Principle

Abstract: One of the 20th century’s most famous local-global theorems is due to Hasse, who proved that for a cyclic extension L/K of number fields, an element of K is a global norm if and only if it is a local norm everywhere. This is no longer true (in general) if the extension L/K is abelian (and non-cyclic), but the question remains: for which extensions L/K of number fields is it true that an element of K is a global norm if and only if it is a local norm everywhere? In this talk, we will give an account of this problem, focusing on history, motivation from geometry, and techniques from finite group theory which have led to some recent progress.