Title: The monoid of product-one sequences over subsets of groups

Abstract: If *G* is a multiplicatively written group and *H* is a subset of *G*, then one can consider the set of all finite formal sequences over *H *up to permutation with concatenation of sequences as operation, such that for each sequence there exists a permutation of the elements whose product is 1. It is called the monoid of product-one sequences over *H* and we will investigate some algebraic and arithmetic properties of it, with a special emphasis on subsets of the infinite dihedral group. This is joint work with Qinghai Zhong.

*Abstract*: A left Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[…[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is a left Engel element precisely when we can choose ${\mathscr E}(g)={ 1}$.) We prove that if a finite group $G$ admits an automorphism $\varphi $ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(\varphi )$ has a left Engel sink of cardinality at most $m$, then the index of the second Fitting subgroup $F_2(G)$ is bounded in terms of $m$.

A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[…[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)={ 1}$.) We prove that if a finite group $G$ admits an automorphism $\varphi $ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(\varphi )$ has a right Engel sink of cardinality at most $m$, then the index of the Fitting subgroup $F_1(G)$ is bounded in terms of $m$.

]]>The preprint concerns infinite-dimensional Lie algebras, over a field of prime characteristic *p*, which are graded over the positive integers and have maximal class in the sense that the *k*th Lie power of *L* has codimension *k* in *L*, for all *k>*1. The *type p* qualifier refers to the fact that the focus is on those generated by an element of degree *p*, together with an element of degree 1. (The actual natural definition is a bit more restrictive.) When the prime *p* equals 2, such Lie algebras were completely classified by Caranti and Vaughan-Lee in 2003. An extension to odd primes *p* presents many challenges which were addressed in Claudio’s PhD thesis (Trento, 2014). Claudio achieved a classification but at the expense of quite intricate calculations. Turning some of those calculations into more readable arguments was done by Valentina in a portion of her PhD thesis (Lincoln, 2019), and required producing new ideas of wider applicability. Finally, further simplification work, and addition of new results, were done by the authors through an online collaboration over several months.

The 40-page preprint includes several main parts of the classification proof, but not all, and a separate paper is planned for completing its exposition. A generalization to *type n*, meaning still in characteristic *p* but generated by an element of degree 1 and one of arbitrary degree *n*, is a bigger challenge. Only the special cases *n*=1, 2, *p* have been settled. Some preliminary steps in the general case were taken by Sandro’s former student Simone Ugolini in his PhD thesis (Trento, 2010), and we hope that those results will also eventually find their way to publication. We also believe that some of the techniques developed in our recent preprint will allow progress on the general case of type *n*.

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

All talks will be held online via Microsoft Teams.

The timetable is as follows, where all times are given in Greenwich Mean Time:

13.15-14.15: Peter Kropholler (University of Southampton), talk title to be confirmed

14.20-15.20: Ilaria Castellano (University of Milano-Bicocca), talk title to be confirmed

15.20-15.50: tea and coffee break

15.50-16.50: Nansen Petrosyan (University of Southampton), talk title to be confirmed

More details to be found at https://www.lancaster.ac.uk/maths/fcg/

]]>Title: Using the Classification of Finite Simple Groups to determine quotients of Aut(F_n) and Mapping Class Groups.

Abstract: This is joint work with Barbara Baumeister and Dawid Kielak. As the title suggests, we show how the CFSG can be used to exclude various (infinite) families of finite simple groups as quotients of SAut(F_n) (n \geq 3) and the Mapping Class Group of a surface of genus at least 3.

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