University of Manchester, Wednesday May 23rd 2018
Venue: Room G108 of the Alan Turing Building, University of Manchester.
Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine – note that most people in this category are elgibile for a rail-card).
13:00-13:50 Raul Moragues Moncho (University of Birmingham) Fusion systems over p-groups with an extraspecial subgroup of index p
Saturated fusion systems are constructions on p-groups which generalise the action of a finite group on its Sylow p-subgroups via conjugation. They have uses in representation theory and topology, but we focus on finite simple groups. Starting from a p-subgroup, we introduce homomorphisms between its subgroups which behave like conjugation by an element of an overgroup. There arise exotic configurations which do not come from finite groups, one of which was found in the search for the finite simple groups.
In order to find and study new exotic configurations, fusion systems have been classified over certain families of p-groups. We focus on p-groups S with an extraspecial subgroup of index p when p is odd, which appear as Sylow p-subgroups of PSL4(p), PSU4(p), PSp4(p) and G2(p) if p is large enough, as well as some other finite simple groups when p is small. We present a classification of fusion systems over S with no normal p-subgroups, which includes some new exotic examples related to a family constructed by Parker and Stroth.
It is well known that every finite simple group can be generated by two elements. Moreover, two arbitrary elements are very likely to generate the entire group. For example, every non-identity element of a finite simple group belongs to a generating pair. Groups with the latter property are said to be 3/2-generated. It is natural to ask which other finite groups are 3/2-generated. In 2008, Breuer, Guralnick and Kantor conjectured that a finite group is 3/2-generated if and only if every proper quotient of the group is cyclic.
In this talk, I will introduce this area of research before reporting on recent progress towards establishing this conjecture, where probabilistic techniques play a key role. I will also highlight connections with related topics, such as recent work on generating graphs.
Artin groups are those groups with presentations of the form
< x1, x2, …, xn | xixjxi…= xjxixj… where each word has length mij, 0<i,j<n+1>
where mij are at least 2 and possibly infinite. Unless an Artin group is free, it
can’t be hyperbolic because it contains subgroups isomorphic to the free abelian group of rank 2. But despite that, it’s been known for more than 30 years that many Artin groups (certainly all those of ‘large’ type, for which no pairs of generators commute, or rather, mij>2, for all i,j) exhibit a weak form of relative hyperbolicity. Recent work of Huang and Osajda now proves that all those Artin groups, and a few more, are ‘systolic’, that is, they act geometrically on ‘systolic complexes’. Hence, in particular, they are all biautomatic, and display various algorithmic properties which are always found in hyperbolic groups.
This is consistent with my own relatively recent work with Holt, which proved that all these Artin groups (and more) are shortlex automatic, with effective rewrite systems to a (shortlex minimal) normal form. I’ll talk about ongoing work with Holt, to try and explain how these quite different approaches give many of the same answers.