Title: Automorphism groups, elliptic curves, and the PORC conjecture

Abstract:

In 1960, Graham Higman formulated his famous PORC conjecture in relation to the function f(p,n) counting the isomorphism classes of groups of order p^n . By means of explicit formulas, the PORC conjecture has been verified for n < 8. Despite that, it is still open and has in recent years been questioned. I will discuss (generalizations of) an example of du Sautoy and Vaughan-Lee (2012), together with a conceptualization of the phenomena they observe. Hidden heroes of this story turn out to be Hessian matrices and torsion points of elliptic curves. This is joint work with Christopher Voll.

Our speakers were Peter Kropholler (University of Southampton), Ilaria Castellano (University of Milano-Bicocca) and Nansen Petrosyan(University of Southampton).

]]>Title: New Periodic Spinal Groups

Abstract: We introduce a very natural class of spinal groups mimicking the Gupta–Sidki 3-group, and establish a condition for these groups to be periodic. More refined results on periodicity are discussed, showing that almost every finite group may occur as a rooted group of a periodic spinal group.

New paper by **Evgeny Khukhro (Univ. of Lincoln) and Pavel Shumyatsky (Univ. of Brasilia)** “Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks” has been accepted for publication in the *Monatshefte für Mathematik*. The work was supported by FAPDF and CNPq-Brazil, and stems from the collaboration with University of Brasilia.

*Abstract*: A right Engel sink of an element of a group is a set such that for every all sufficiently long commutators belong to . (Thus, is a right Engel element precisely when we can choose .) We prove that if a profinite group admits a coprime automorphism of prime order such that every fixed point of has a finite right Engel sink, then has an open locally nilpotent subgroup.

A left Engel sink of an element of a group is a set such that for every all sufficiently long commutators belong to . (Thus, is a left Engel element precisely when we can choose .) We prove that if a profinite group admits a coprime automorphism of prime order such that every fixed point of has a finite left Engel sink, then has an open pronilpotent-by-nilpotent subgroup.

]]>This year’s British Mathematical Colloquium was supposed to be held at the University of Glasgow, but because of COVID restrictions it was held virtually in Glasgow. The organisers did an amazing job of making attendees feel like they were actually in Glasgow, using a platform called Sococo. You could “walk” around a top-down view of buildings at the university. Talks were held in various virtual rooms, so your avatar would enter a room to see the link to the zoom lecture taking place in that room. Sococo allowed the organisers to put a beach near the campus so attendees could soak up some virtual sun.

Simon Smith attended the conference, and gave an invited talk in the Group Theory workshop on his recent joint work with Colin Reid on the theory of local action diagrams (essentially a “local action” complement to classical Bass-Serre theory). Other speakers (who were all excellent) in the workshop were:

- Robert Chamberlain (Warwick), who spoke about minimal permutation representations of finite groups;

- Nayab Khalid (St Andrews), who spoke about an infinite geometric presentation for Thompson’s group F;

- Alan Logan (Heriot-Watt), who spoke about equalisers of free group homomorphisms and Post’s correspondence problem;

- Radhika Gupta (Temple), who spoke about non-uniquely ergodic arational trees in the boundary of Outer space;

- Alex Evetts (ESI), who spoke about conjugacy growth of finitely generated groups; and

- Gareth Tracey (Oxford), who spoke about Invariable generation and the Chebotarev invariant of a finite group

This Regional Meeting will be followed by a 2-day **workshop on Profinite Groups and Related Aspects**. The speakers for the workshop include Gunnar Traustason (University of Bath), Anastasia Hadjievangelou (University of Bath), Nadia Mazza (Lancaster University), John Wilson (University of Cambridge and University of Leipzig), Pavel Shumyatsky (University of Brasilia), Colin Reid (University of Newcastle, Australia), Alejandra Garrido (Universidad Carlos III de Madrid), Henry Bradford (University of Cambridge), and Rachel Camina (University of Cambridge).

More details can be found here: https://profinitelincoln2021.wordpress.com/

This meeting is supported by the LMS Regional Meeting Grant, and by the School of Mathematics and Physics of the University of Lincoln.

**Ischia Group Theory 2020/2021**”, which will now take place online on 25-26 March 2021 (originally planned to be held in Ischia (Italy), from 30th of March to 4th of April, 2020, but postponed due to the pandemic). This biannual conference is one of the major events in the academic calendar attracting experts in group theory from all over the world.