**Anitha Thillaisundaram** (Lund University) will talk about her work on the **17th of June** in INB3235 at** 3pm.**

Everyone is welcome to attend.

Title: The Amit-Ashurst conjecture for finite metacyclic p-groups

Abstract: The Amit conjecture says that for a word map on a finite nilpotent group G, the probability of the identity element occurring in the image of the word map is at least 1/|G|. This conjecture has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring in the image of a word map on a finite nilpotent group G is either 0, or at least 1/|G|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit-Ashurst conjecture and show that the Amit-Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup.

]]>*Abstract*: In the category of locally compact Hausdorff groups LCG one has to distinguish between algebraic morphisms and algebraic and continuous morphisms. Let Epi(L,G) be the set of surjective group homomorphisms and cEpi(L,G) the subset consisting of continuous surjective group homomorphisms. The question we address is the following: Under which conditions on the discrete group G does the equality Epi(LCG,G)=cEpi(LCG,G) hold? In particular, we show that any surjective group homomorphism from a locally compact Hausdorff group into the automorphism group of a right-angled Artin group Aut(A_\Gamma) is continuous.

Congratulations to our Year 4 Maths Masters student **Thomas Smith** with getting a PhD position at the Heinrich Heine University in Düsseldorf! Building on excellent result of his mathematics studies in Lincoln, including the 2021 summer UROS research project “Engel conditions in branch groups” under supervision of Dr Anitha Thillaisundaram, Thomas made further progress in his year 4 studies, which in the 2nd semester amounts to a massive quadruple module of Masters research project (title: “Groups of automorphisms and their actions on regular rooted trees”, supervisor Prof. Evgeny Khukhro). Thomas’ successful application for PhD studies in Düsseldorf means that in September of 2022 he will start working on his PhD thesis under supervision of Prof. Benjamin Klopsch, Head of the Algebra and Number Theory group at the HHU Düsseldorf, in the research area of Asymptotic and Geometric Group Theory. Well done, Thomas, and best wishes in your further studies and research!

Title: Flag Hilbert-Poincaré series and related zeta functions

Abstract: We define a class of multivariate rational functions

associated with hyperplane arrangements called flag Hilbert-Poincaré

series. We show how these rational functions are connected to Igusa zeta

functions and class counting zeta functions for certain graphical group

schemes studied by Rossmann and Voll. We report on a general

self-reciprocity result and explore other connections within algebraic

combinatorics.

This is joint work with Christopher Voll and with Lukas Kühne.

*Abstract*: We prove several structural results on *Nottingham algebras*, a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called *diamonds*. The first diamond occurs in degree 1, and the second occurs in degree *q*, a power of the characteristic. Each diamond past the second is assigned a *type*, which either belongs to the underlying field or is ∞. Nottingham algebras with a variety of diamond patterns are known. In particular, some have diamonds of both finite and infinite type. We prove that each of those known examples is uniquely determined by a certain finite-dimensional quotient. Finally, we determine how many diamonds of type ∞ may precede the earliest diamond of finite type in an arbitrary Nottingham algebra.

Title: When are two elements conjugate?

Abstract: understanding the structure of conjugacy classes is essential in the study of a group. We will see how the conjugacy classes of a group can be understood using the group action. We will analyze the conjugacy classes for a variety of interesting examples, including groups acting on trees and group almost acting on trees, following a joint work with Waltraud Lederle. On the way, I will give a short intro to Neretin’s group: the group of almost automorphisms of a regular tree.

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