Inna Capdeboscq started proceedings this morning with a talk about one small part of the second-generation (Gorenstein–Lyons–Solomon) proof of the Classification of Finite Simple Groups. What she described will be volume 8 of the complete proof, if I remember correctly. All I can really say is that I take my hat off to the heroes who are doing this job; I would be completely incapable of doing it myself. But we owe these heroes a great debt.
The product replacement algorithm for choosing a random element of a finite group G was invented by my Queen Mary colleagues Charles Leedham-Green and Leonard Soicher. It works as follows. You choose n sufficiently much larger than the number of elements required to generate the group, and pick a starting n-tuple which generates the group. Each step in the random walk requires a choice of distinct i and j and two further random bits; element number i of the tuple is multiplied, either on the left or on the right, by either element number j or its inverse. After sufficiently many steps, you read off the first element of the tuple.
There is some controversy about whether the distribution of the n-tuple converges to uniform on the set of generating n-tuples, and if so, how fast. But Laci Babai and Igor Pak pointed out another problem. The projection of the uniform distribution on generating n-tuples onto the first component may not give the uniform distribution on elements of G. They gave examples to show that this can happen. Andrea Lucchini showed that, even for soluble groups, this problem is unavoidable.
For me the highlight of the day was Aner Shalev’s talk. Most of it was devoted to old and new results about fixed points of elements of a permutation group, base size, proportion of fixed-point-free elements, and invariable generation of finite groups. Beautiful stuff, and my name is attached to some of the conjectures that are now proved. But I liked best the end, when Aner turned to infinite groups.
A subset S of a group G invariably generates G if, when each element of S is replaced by an arbitrary conjugate, the resulting set generates G. There are some recent and powerful results, some of which were described by Eloisa Detomi yesterday: for example, Aner told us that every non-abelian finite simple group has a 2-element invariable generating set, a really powerful improvement of the fact that every such group has a 2-element generating set.
But, in the infinite case, it can happen that G doesn’t invariably generate itself! The easiest example is the general linear group over the complex numbers: by elementary linear algebra, every invertible matrix is conjugate to an upper triangular matrix (in Jordan form). This example is familiar in another context: the general linear group, acting on the cosets of the group of upper triangular matrices, has no fixed-point-free elements. Aner pointed out the simple fact that the two things are equivalent: a group invariably generates itself if and only if it has the Jordan property (every non-trivial transitive action has fixed-point-free elements). He gave the name IG-groups to the class of groups with this property, and asked for a characterisation of this class of groups.
It rained (gently) all day, so I didn’t leave the Rényi Institute until after the last talk. We are promised an improvement in the weather (fortunately). This morning I saw a fire engine pumping water out of a building; last night, someone was emptying his basement with a bucket chain.