Groups, geometries and representations

Here, more than a week late, are comments on the conference formally titled “Groups, Geometries and Representations, Oxford 2018”, actually a birthday conference for Dan Segal and Aner Shalev.

I stayed in Balliol College’s new Jowett Walk Building, just around the corner from Holywell Manor, the Balliol – St Anne’s graduate centre where I stayed as a graduate student nearly 50 years ago. So I was able to walk through the University Parks to the Mathematical Institute as I did back then, although the Institute itself has moved to a new site where the Radcliffe Infirmary once stood.

As usual, I can only highlight a few talks. On the firt morning, Bob Guralnick told us about the discovery of some huge first cohomology groups of finite groups which led to the demise of Tim Wall’s conjecture (that the number of maximal subgroups of a finite group did not exceed the order of the group). Actually my reading of the write-up of this suggests that Wall’s conjecture is very nearly true, and the number of maximal subgroups of a group of order n may be bounded by n1+ε for some small ε.

Martin Bridson started his talk with a little gem. Consider the three groups with presentations as follows:

  • G2 = ⟨a,b:ba=a2b
  • G3 = ⟨a,b,c:ba=a2b,cb=b2c,ac=c2a
  • G4 = ⟨a,b,c,d:ba=a2b,cb=b2c,dc=c2d,ad=d2a

Then he said, one of these groups is infinite with infinitely many finite quotients; one is infinite with no non-trivial finite quotients, and one is trivial. He took a vote on which is which, which was indecisive; this reinforced his point that a presentation is a very bad way to convey information about a group.

I happened to know the answer. G2 clearly has many finite quotients, since all we need is a group with an element conjugate to its square. (In fact this is the Baumslag–Solitar group. I recognised G4 as a group used by Graham Higman exactly because it is infinite with no non-trivial finite quotients (a pleasant exercise), and I had once used it for a similar purpose. Therefore G3 must be trivial. (Is this a proof of that fact??)

He went on to speak about what information the set of finite quotients of a group (in other words, its profinite completion) tells about the group (assuming it to be residually finite, unlike Higman’s group, which is not distinguished from the trivial group by its finite quotients).

On the second day, John Thompson talked about projective planes. He had some difficulty in writing on the board, but he spoke completely clearly. He presented us with an outrageous conjecture, designed to challenge us to solve the extremely difficult problem of finding the set of orders of finite projective planes. His conjecture was that this set is equal to the set of orders of direct powers of finite simple groups, both abelian (which gives the familiar prime powers) and non-abelian (so the first test case would be 60). In the title of the talk he mentioned the Hall–Paige conjecture; he ran out of time to tell us about this conjecture and its proof, but did so in answer to a question.

On Wednesday, Alex Lubotzky started his talk by saying, “I should thank the organisers for inviting me. But I found out later from Martin Liebeck that I was an organiser.” Indeed, organisation was very low-key. There was no paper version of the programme or abstracts, or even paper to write on; and registration consisted of participants sorting through a random heap of name badges on a table to find their own. (I am sufficiently old-fashined that I like having at least a printed programme. Indeed, when Martin asked me to chair a session, I said that my fee was a hard copy of the programme.)

Alex proceeded to give a beautiful talk, one of the highlights of the conference. He used the phrase “Finite to infinite”. This reminded me of an evening in a restaurant in Budapest, where Laci Babai and I planned a conference with the title “Group Theory: Finite to Infinite”; it was held in a hotel near Pisa, where there was much discussion of profinite groups, and where we shared the hotel with a football team (I don’t now recall which one) whose manager was constantly on the phone trading players and doing other deals.

There was much in Alex’s talk, but here is just one thing, which I had heard about but not internalised before. This is the notion of a sofic group, one which can be approximated by finite symmetric groups with normalised Hamming distance. (This means that there are near-homomorphisms from G to these groups, where the images of any non-identity element don’t get arbitrarily close to the identity.) Of course, every finite group can be embedded into a finite symmetric group; but infinite groups are much wilder. Nevertheless, it is currently open whether every group is sofic. If so, this would be an extraordinary statement of the power of the innocent-looking group axioms; so, if I had to bet, I would put money on the existence of non-sofic groups. But it appears to be a very hard question.

I can’t fail to mention Goulnara Arzhantseva’s construction of expander graphs where the diameter to girth ratio is bounded, from high-dimensional linear groups.

Nor can I fail to mention Persi Diaconis, was doing random walks on the irreducible characters of finite or compact groups, in a way which recalled the McKay correspondence. You pick a fixed irreducible χ; if you are currently at a character φ, you tensor φ with χ, decompose the result into irreducibles, and pick an irreducible with probability proportional to the product of its degree and its multiplicity. Many familiar and unfamiliar random walks arise. For example, for SU(2), you get the usual random walk on the integers conditioned on never going negative. He also developed a modular version of the same thing.

Cheryl Praeger talked about finding combinatorial structures whose automorphism groups are maximal subgroups of finite symmetric groups. For intransitive groups, you take a subset; for imprimitive groups, a partition (a set of subsets, pairwise disjoint and covering all points); for primitive non-basic groups, a Cartesian structure (which, as Laci Kovács pointed out, can be regarded as a set of partitions satisfying appropriate conditions). For affine groups, you take the affine space. What about diagonal groups with more than two factors? Cheryl described these as collections of Cartesian structures. It seems to me, at least for three factors, that they can be described as Latin squares, essentially the Cayley tables of the factors. (Most naturally, the diagonal group preserves the set of triples whose product is the identity.) This story is not yet finished.

Katrin Tent showed us the construction of infinite “non-split” sharply 2-transitive groups, at least for characteristic 0 or 2. For finite odd characteristic, the problem is more difficult. But she claims that the methods they have used to handle this case may actually lead to substantial advances on the Burnside problem, at least for odd exponent. Again, the story is not finished.

Colva Roney Dougal (the latest of my former students to be made a Professor) opened the proceedings on Friday with a talk entitled “Random subgroups of finite symmetric groups”. The main thrust is the new results she and Gareth Tracey have almost finished on this problem where you choose from the uniform distribution on subgroups, but she began with a beautiful summary on other methods such as random generation. Yet another not-quite-finished story; the precise result is not available yet and is eagerly awaited.

Colva also revealed that Friday happened to be Cheryl’s birthday.

The conference was concluded by a lovely talk from Ben Green. He had proved this theorem as a result of a question from Kufei Zhao, but he didn’t tell us about that. His result says that symmetric subsets of the unit sphere (that is, orbits of a finite group) in high dimensions are almost flat, in the sense that they lie on a thin slice through the centre of the sphere (precisely, there is a unit vector whose inner products with vectors in the symmetric set are in absolute value bounded by c/√(log d) in dimension d. He started by “cooking” his own theorem by pointing out that this is not as surprising as it seems. The vertices of the cube, for example, normalised to lie on the unit sphere, have all coordinates ±1/√d, and so are almost orthogonal to a unit basis vector.

The theorem itself used a quantified version of the theorem of Jordan, according to which a finite subgroup of the unitary group U(d) has an abelian subgroup bounded by a function of d. The quantification was due to Michael Collins (I believe Boris Weisfeiler also worked on this problem but his results were unpublished). Anyway, Collins’ result was not quite strong enough to use as a black box by Green, so he had to open the box and re-do some of the arguments. Not a straightforward task, and a number of ideas from probability theory also crept in.


About Peter Cameron

I count all the things that need to be counted.
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2 Responses to Groups, geometries and representations

  1. Gordon Royle says:

    Do you know what considerations have led John Thompson to his projective plane conjecture? It would be quite spectacular if true, but no matter how hard one wishes something were true, the universe only very occasionally obliges.

    • I agree with you, but I know no more than I said above: he wanted to give people a challenge to work on the problem. I suppose disproving a conjecture of John Thompson would be a feather in someone’s cap.

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