Abstracts

Tim Burness

Title: The regularity number of a finite group and other base-related invariants

Abstract: Let G be a transitive permutation group on a finite set X, let H be a point stabiliser and recall that the base size of G, denoted b(G,H), is the minimal size of a base for G. Equivalently, the base size is the minimal integer k such that G has a regular orbit on the Cartesian product (G/H)^k. As a natural generalisation, let us say that a k-tuple (H_1,..., H_k) of core-free subgroups of G is regular if G has a regular orbit on G/H_1 x ... x G/H_k. Then the regularity number of G, denoted R(G), is the minimal integer k such that every k-tuple of core-free subgroups of G is regular. More refined versions can be defined by imposing additional conditions on the component subgroups, such as solubility or nilpotency, and this leads to natural generalisations of several widely studied conjectures on bases due to Cameron, Pyber and Vdovin.

In this talk, I will introduce the main definitions and I will discuss some of the probabilistic and computational methods we use to study the regularity of subgroup tuples. I will conclude by presenting several new results in the almost simple setting, which extend some of my earlier work on bases with Guralnick, O'Brien, Saxl and Wilson.

This is joint work with Marina Anagnostopoulou-Merkouri.

Cheryl Praeger

Title:   A collegial friendship across the globe: group actions from Oxford to Warwick and Perth 

Abstract: Derek Holt and I were DPhil students together in Oxford, and our mathematical paths have criss-crossed over the decades. I will reflect on our mathematical interactions and finish by describing one of my current research projects where Derek’s work is proving invaluable. It involves absolutely irreducible actions of finite simple groups in small dimensions, and it underpin new procedures to compute with finite classical groups.

William Kantor

Title:  Some groups of order n^3.

Abstract:  I will discuss the relationship between some groups and a standard combinatorial conjecture.

Alex Evetts

Title: Twisted conjugacy growth of virtually nilpotent groups

Abstract: The conjugacy growth function of a finitely generated group is a variation of the standard growth function, counting the number of conjugacy classes intersecting the n-ball in the Cayley graph. The asymptotic behaviour is not a commensurability invariant in general, but the conjugacy growth of finite extensions can be understood via the twisted conjugacy growth function, counting automorphism-twisted conjugacy classes. I will discuss what is known about the asymptotic and formal power series behaviour of (twisted) conjugacy growth, in particular some relatively recent results for certain groups of polynomial growth (i.e. virtually nilpotent groups).

Tara Macalister Brough

Title: Preserving self-similarity in free products of automaton semigroups

Abstract: The concept of self-similarity, famous source of interesting examples in group theory, extends naturally to semigroups.  A self-similar semigroup can be viewed as the semigroup generated by the states of an automaton (more formally, a deterministic synchronous - but not necessarily finite - transducer) acting on strings over a finite alphabet.  If the automaton is finite, we have an automaton semigroup.

We examine the extent to which the classes of self-similar semigroups and automaton semigroups are closed under the free product operation, substantially improving on earlier work by Brough and Cain.

This is joint work with Jan Philipp Wächter and Janette Welker.

Susan Hermiller

Title: Subgroups of the group of dyadic piecewise linear homeomorphisms of the real line

Abstract: The group of dyadic orientation-preserving piecewise linear (PL) homeomorphisms of the unit interval is called Thompson's group F, and the question of which groups are - or cannot be - subgroups of F has yielded many interesting results. In this talk I'll discuss the question of what groups can or cannot be subgroups of Aut(F) (the automorphism group of F), and more particularly subgroups of an index 2 subgroup of Aut(F) that is isomorphic to a group of dyadic PL homeomorphisms of the real line. This is joint work with Conchita Martinez-Perez.

Marston Conder

Title: Some personal and professional perspectives on Derek Holt and his achievements

Abstract: Derek was my acting supervisor for the first year of my doctorate at Oxford (while Graham Higman was away for a year on sabbatical). Derek was very very helpful and I owe him a huge debt of gratitude for his quietly enabling me learn a lot of group theory from him.  Over the years since then, I have learnt even more (and benefitted) from a wide variety of mathematical contributions by Derek -- made using his extensive knowledge of group theory, linear algebra, combinatorics, representations, cohomology, computational languages and systems and other things. In this talk I'll describe just a few of those contributions, especially concerning finitely-presented groups and algorithms for gaining important information about them, and some significant outcomes of their application.

Bettina Eick

Title: Computing with polycyclic groups: new developments and open problems

Abstract: This talk gives a brief introduction into the algorithmic theory of polycyclic groups with a view towards {\em practical} algorithms. It then considers some new developments: one is the computation of the Frattini subgroup of a polycyclic group, the other is a solution to the isomorphism problem and the computation of genera for finitely generated torsion free nilpotent groups of small Hirsch length. The talk provides an overview on these two new developments and it finishes with some still open problems.

Saul Schleimer

Title: Solving the word problem in the mapping class group in quasi-linear time.

Abstract: The word problem for the mapping class group was first posed, and first solved, by Dehn [1922] in his Breslau lectures.  His method was rediscovered, and greatly extended, by Thurston [1970-80's].  Mosher [1995] proved that the mapping class group is automatic and so found a quadratic-time algorithm for the word problem.  Hamidi-Tehrani [2000] and Dynnikov [2023] gave quadratic-time algorithms using train-tracks.

We give the first sub-quadratic-time algorithm.  We combine the work of Dynnikov with a generalisation of the half-GCD algorithm to obtain an algorithm running in time $O(n \log^3(n))$.

This is joint work with Mark Bell.