Group Theory Day in Budapest

Péter Pál Pálfy (P cubed to his many friends) is 70; his birthday was celebrated by a Group Theory Day at the Rényi Institute in Budapest.

The morning was devoted to talks by three of P cubed’s students. First off was Miklós Abért, telling us about random generation of finite or infinite groups. Sometimes random methods work well, sometimes things are much harder; the distinguishing thing seems to be how much “randomness” there is available. If you take the rooted binary tree and randomly flip the two branches at each vertex, the orbit tree of the resulting group is a Galton–Watson tree, and everything is well known. But if you choose k random rotations of the 2-sphere, it is not known whether the group you get has a spectral gap almost surely or not. Miklós thinks so, but cannot (yet) prove it.

Second, after coffee, was Zoltán Halasi, on the diameters of groups. The diameter of a group is the maximal diameter of the Cayley graph of the group with respect to any symmetric generating set. Babai’s conjecture (still not proved) is that the diameter of a simple group is of order a polynomial in the logarithm of the group order. He described some of the results for finite classical groups, using some hard results.

The third was Balázs Szegedy, on his quest for higher-order representation theory. I have to say that at the end of the talk I am not sure what would count as success in this quest; what he is looking for is not higher category theory, though he feels that there must be a connection. It is related to the hypergraph regularity lemma, higher-dimensional Fourier analysis, and the structure theory of characteristic factors in dynamics, and will almost certainly have innteresting applications.

After a nice lunch in a Sicilian restaurant, it was my turn. I talked about inverse group theory, which I have spoken about elsewhere; but I added two topics, one old, one new, at the end. Why should inverse group theory be restricted to functions from groups to groups? Two constructions of combinatorial objects from groups are Cayley tables and commuting graphs. For Cayley tables, Michel Frolov (a French army officer who was apparently unaware of what his countryman had done on group theory) gave a condition which recognises Cayley tables of groups among Latin squares, the quadrangle condition. For the commuting graph, the paper by Arvind, Ma, Maslova and me (already on-line in the Journal of Algebra) gives a quasi-polynomial recognition algorithm; we do not expect there to be a really simple characterisation.

The few days before and after the event gave a chance to revisit some of my favourite parts of one of my favourite cities, including Margaret Island, the Matthias church and Fisherman’s Bastion, the Chain Bridge, the market hall, and music beside the Danube. But one new find, just around the corner from the Rényi Institute, is the Cat Museum, full of paintings, cartoons, photographs, and postcards of all things feline, including Kim Jong Un explaining how “In my country, we love cats!” There was also time to spend with P cubed, Miklós, Gabor Kun, and Csaba Szabó.