I wouldn’t usually treat the sender of such an email like this, but I happened to notice that all the other addressees had names beginning `pj`, so I doubt that I had been carefully selected.

A glance at the attachment showed two things.

- First, it contained two proofs that π = (14−√2)/4.
- Second, it was not a manuscript or preprint, but a reprint from an international journal with the spuriously precise impact factor of 3.785.

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I already posted part of this picture, a signpost on the edge of the Olympic Park back in 2011. I tagged it with the last phrase of the second verse of Procol Harum’s apocalyptic song *Homburg*: “The signposts cease to sign”.

I decided that, instead of scrubby trees by an east London waterway, it deserved a more dramatic background, so I used a photo of the sun setting into the sea taken at St Kilda, a suburb of Melbourne, later that year.

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The picture above is not friendly team competition over dinner at a sandpit, but shows the kings of Hungary, Bohemia and Poland meeting in the castle of Visegrád in 1335 to hammer out an agreement. Did the agreement include relaxing the Hungarian quota on imports of Czech beer?

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Szentendre is a small town at the end of a suburban railway line from Budapest. I first visited it 22 years ago, in the winter; it was a beautiful place of artists’ studios and galleries, old Serbian churches full of icons, and wide views across an arm of the Danube to a large island.

We went back yesterday, on a summer Sunday at the end of the St Stephen’s Day long weekend, and could hardly move for tourists.

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The moment of the rose and the moment of the yew tree

Are of equal duration

T. S. Eliot, “Little Gidding”

This picture is of a corner of the sadly neglected gardens around the tomb of Gül Baba, the person who introduced the cultivation of roses to Budapest.

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Now the conference is over.

On the last morning, Marty Isaacs posed an interesting problem. Let *p* be a prime, and *n* a positive integer. Is there an infinite group in which exactly *n* elements are not *p*th powers? This was during a talk in which he told us several things about finite groups with this property, with complete proofs: notably, either such a group has order at most *n*^{2}, or it is a sharply 2-transitive group of order *n*(*n*+1), where *n* is a power of *p*.

I gave the last talk of the meeting. Contrary to my usual practice, I will tell you a bit about it, for two reasons. First, this is joint work with Colva Roney-Dougal; we don’t have a preprint yet, but I regard it as public following my talk, and so worth publicising. The second reason will emerge later.

The aim of the project is to say something about generating sets for finite groups. In the case of an elementary abelian *p*-group, the minimal generating sets are just bases in a vector space, and the theory is so easy that we teach it to all mathematics students. By the Burnside basis theorem, arbitrary *p*-groups are essentially no worse. A set generates a *p*-group *P* if and only if the corresponding cosets generate *P*/Φ(*P*), an elementary abelian group (where Φ(*P*) is the Frattini subgroup); so we just have a vector space “blown up” by the order of the Frattini subgroup. But for arbitrary groups, things are more complicated.

One gadget which has been much studied recently is the *generating graph* of a group *G*, in which group elements *x* and *y* are joined if and only if {*x,y*} = *G*. Of course, this is the null graph unless *G* is 2-generated; but, by a miracle which we don’t understand, at least every finite simple group is 2-generated.

When I first learned about this, I calculated the generating graph of *A*_{5}, a group whose order is equal to the age of P-cubed. I found to my astonishment that the automorphism group of the graph has order 23482733690880. But it is not a new simple group; the reason is not hard to find. Elements of order 3 or 5 which generate the same cyclic subgroup have the same neighbour set in the graph, and so can be permuted arbitrarily. There is a normal subgroup of order (2!)^{10}(4!)^{6} doing this, and the quotient is just *S*_{5}, the automorphism group of *A*_{5}.

So we can reduce the graph by the “g-equivalence relation”, in which two vertices are equivalent if they have the same neighbours. The resulting smaller graph shares many important properties (including clique number and chromatic number) with the original one, and is likely to have a more sensible automorphism group.

We were led to another equivalence relation, the “m-equivalence relation”, in which two elements are equivalent if they lie in the same maximal subgroups of the group *G*. It is clear that this is a refinement of g-equivalence, but is non-trivial for any finite group. So now I can state one of our conjectures:

*Conjecture*: In a finite simple group (or maybe even a 2-generated almost simple group), two elements are g-equivalent if and only if they are m-equivalent.

This is false in general, even for 2-generated groups. For *S*_{4}, for example, a double transposition lies in no 2-element generating set, so they are g-equivalent to the identity, but not m-equivalent: there are 14 g-equivalence classes, but 15 m-equivalence classes. The numbers of m-equivalence classes in symmetric groups form an interesting sequence, beginning 1, 2, 5, 15, 67, 362, 1479, …

However, I learned later that Budapest is not a good place for marketing generating sets; they are readily available on street corners:

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Thursday was a national holiday in Hungary, St Stephen’s Day if you are religious or traditional, Constitution Day if you are a communist. “The shrewd communists let the parliament pass the new constitution on August 20, 1949,” in the words of P-cubed, who kindly invited the foreign guests to his apartment, where we had a grandstand view of the fireworks from the Citadel hill. It was a splendid display, with the fireworks from two sites synchronized so as to mirror each other; some of them exploded in the Hungarian colours of red, white and green, and others in many other colours. A smaller display from the Castle was not so easily visible. We were also provided with delicious food and very nice palinka.

Earlier in the day, the fireworks at the conference were provided by Laci Babai, who talked about symmetry and regularity. I have heard versions of this talk before, but it is always good to hear it again, there is always something new. The main thrust is that highly regular objects cannot have too much symmetry, in a suitable asymptotic sense; rather, the regularity works against symmetry in some way.

In 1980, Laci proved his beautiful result that a primitive but not 2-transitive subgroup of the symmetric group *S _{n}* has order bounded by the exponential of the square root of

At about the same time, it was realised that the Classification of Finite Simple Groups (announced then, but not proved for another quarter of a century) gave a substantial strengthening; not in terms of the bound as such, but one could bring the bound down to about *n*^{log n}, or even *n*^{log log n}, by allowing longer and less well-defined lists of exceptions. Laci’s ambition is to prove as much as possible of this strengthening by methods not dependent on CFSG, and recently he and others have had some remarkable successes.

I will just mention one here; he gave us the proof of this in detail. Given a strongly regular graph on *n* vertices which is not trivial (the disjoint union of complete graphs, or the complement) or the line graph of a complete or complete bipartite graph or the complement of this, then the automorphism group of the graph contains no large alternating groups as sections (and hence, if primitive, then its order is subexponential, by the result of the BCP paper). The beautiful proof goes by the following steps (I skip several details):

- Using “elementary number theory”, it is shown that if a permutation in
*S*has order_{n}*n*^{α}, then some power of it moves at most*n*/α points. - A theorem of Seidel asserts that a strongly regular graph with least eigenvalue −2 or greater is one of the excluded examples, or one of finitely many exceptions (which he found explicitly, but which do not affect the asymptotic argument).
- Any other strongly regular graph has least eigenvalue −3 or smaller; this allows spectral techniques to show that the graph is in a weak sense pseudo-random, and the support of an automorphism cannot be too small.
- Combining this with the first step shows that orders of automorphisms are not too large, and hence the degrees of alternating groups appearing as sections are bounded by about (log
*n*)^{2}/(log log*n*).

Another theme of the day was the work of Sasha Borovik and Şükrü Yalçinkaya on a black box recognition algorithm for the group PSL(2,*q*), where *q* is a large prime power. It is important to be able to find a unipotent element, but the probability of finding one by random search is about 1/*q*, which is too small; so great cleverness in the use of involutions is required. From this point, one embeds PSL(2,*q*) into PGL(2,*q*), which is isomorphic to SO(3,*q*), and so acts on the projective plane over the field of *q* elements. The plane can be constructed directly from the group (points correspond to involutions and unipotent subgroups, as do lines), and then the field can be constructed by coordinatisation of the projective plane. I was familiar with the last step, which is classical projective geometry; I worked this out for myself when I wrote my lecture notes on *Projective and Polar Spaces*. Indeed, in my report on computer-assisted finite geometry, which I wrote for John Cannon in 1988, I speculated that computers could carry out this coordinatisation, but I have never seen it done in practice before now!

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A *Ramanujan graph* is a connected finite graph of valency *k* whose eigenvalues (apart from *k* and −*k*) all have modulus at most 2√(*k*−1). This interval is the spectrum of the infinite *k*-valent tree *T* (regarded as an operator on L^{2}(*T*)), and so there is a sense in which a Ramanujan graph is a best finite approximation to the tree; these graphs tend to have high girth and very good expansion properties.

In the late 1980s, Lubotzky, Phillips and Sarnak published a construction of Ramanujan graphs, which was a heady mix of number theory, geometry, representation theory, and combinatorics. I liked the paper so much that I gave a seminar to tell my colleagues about it. I will make a few comments here.

Let *p* and *q* be two primes congruent to 1 (mod 4). In any expression for *p* as a sum of four squares, one is odd and the other three even; if we take the first one to be odd and positive, there are just *p*+1 expressions, by a theorem of Jacobi. We can represent the corresponding 4-tuples as quaternions over GF(*q*), and indeed (by our condition on *q*) as 2×2 matrices. The set of matrices is closed under inversion and generates PGL(2,*q*) or PSL(2,*q*). Now the Cayley graph is the required Ramanujan graph.

From a higher perspective, what is going on here? Take *R* to be either the *p*-adic integers or the formal power series ring over GF(*p*), and *F* its field of fractions. Associated with PGL(2,*F*) and its maximal compact subgroup PGL(2,*R*) is a building, which in this case is the (*p*+1)-valent tree; taking the quotient by a cocompact subgroup gives a graph which is finite (since it is compact and discrete). If the subgroup we choose is the congruence subgroup mod *q*, we get the above construction. Now a theorem of Deligne, off-the-shelf, enables the facts about its spectrum to be proved.

The highlight of yesterday was Alex Lubotzky’s talk. He started out by reminding us of the above. Now he and his colleagues have developed a higher-dimensional version of this. Rather than PGL(2,*F*), they take an algebraic group of higher rank, so that at the first step they get a building of rank greater than 1. Again, the quotient by a congruence subgroup gives a finite quotient. So far, so standard; but they split the adjacency relation in the building into *d* “Hecke-like” operators which, although they are not self-adjoint, are normal (commute with their adjoints) and pairwise commuting, hence are simultaneously diagonalisable, by a theorem which I regard as standard linear algebra (but seems to be less well known than it should be). Now the “spectrum” of the building is a subset of *d*-dimensional space, and contains the essential part of the spectrum of the finite quotient.

They call the resulting object a “Ramanujan complex”. This work was not driven by applications, but for sure applications of such an appealing concept are bound to arise; indeed this process has already started (leaving aside the “building” of that name at IIT Kharagpur in India!)

In the evening, rather than a “conference dinner”, we had a celebration dinner for Péter Pál Pálfy’s birthday. The event began with a concert by a recorder group led by a leading Hungarian musician. They played six pieces, two old (my favourite was Sufi music from Turkey, transcribed in Padova, and now in a manuscript in London), two modern, and two electronic (the first with several mobile phones which had been set to “resonate” with certain notes played by the recorders; in the second, the recorders were not played in the conventional way but controlled the electronics directly). An interesting concert with some really beautiful pieces.

There were many tributes to P-cubed, perhaps my favourite from his supervisor Laci Babai. He explained that Paul Erdős said that, when half your co-authors “leave”, then your time has probably come; but half is measured not by number or even number of papers, but by the “weight” of the work (by which, obviously, he means something different from the number of pages!). For that reason, he (Laci) could not avoid some self-interest in wishing P-cubed a long and prosperous life.

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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.

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The conference opened with a talk by Yoav Segev on his construction, with Eliahu Rips and Katrin Tent, of infinite non-split sharply 2-transitive groups.

A permutation group is sharply 2-transitive if any pair of distinct elements of the domain can be mapped to any other such pair by a unique element of the group. All such finite groups are known, and indeed it is easy to prove that such a finite group is “split”, that is, has a transitive abelian normal subgroup (and hence is the one-dimensional affine group over a finite nearfield – all finite nearfields were determined by Zassenhaus in the 1930s). It is a long-standing open problem whether there exist non-split infinite sharply 2-transitive groups.

Sharply 2-transitive groups give rise to a special low-rank class of independence algebras, a topic that Csaba Szabó and I worked on during my first visit to Budapest more than 22 years ago (at a time when the third Macdonalds had only just opened in Budapest). So it was very interesting to me to hear about this new construction.

The construction is remarkably easy. Yoav took us through the entire thing, apart from some calculations with normal forms in HNN extensions and free products. But the examples seem to be very diverse. In fact they show that any group whatever can be embedded as a subgroup in a sharply 2-transitive group.

In a sharply 2-transitive group, any two points are interchanged by a unique element, which is an involution; all involutions are conjugate, and so all fix the same number of points, which is either 0 or 1. Their construction deals with the former case (which they call “characteristic 2”). It can be formulated in group-theoretic terms. The final result *G* is a group with a subgroup *A* having the properties that any two conjugates of *A* intersect in the identity, that there are only two *A*–*A* double cosets in *G*, and that *A* contains no involutions. (*A* is the point stabiliser.) So start with any pair *G*_{0}, *A*_{0} having the first and third properties; if there are more than two double cosets, add a new generator to unify two of them. This may create new double cosets, but “in the limit” (repeating the construction enough times) the tortoise catches up with the hare and all the double cosets outside *A* are pulled into one. Now to get the announced result, we can take *G*_{0} to be any group and *A*_{0} to be the trivial group.

Balázs Szegedy gave us a very interesting talk on how nilpotent groups force themselves into additive combinatorics (e.g. Szemerédi’s theorem on arithmetic progressions in dense sets of integers) whether we like it or not. Roth proved the theorem for 3-term arithmetic progressions using Fourier analysis; a similar proof of Szemerédi’s theorem involves “higher order Fourier analysis”. Balázs has axiomatised the appropriate objects (which he calls *nilspaces*) in terms of “cubes”. The axioms are simple enough but the objects captured include nilmanifolds. I cannot really do justice to the talk in a single paragraph; but the claim is that shadows of these objects already appear in Szemerédi’s proof (although it appears to be just complicated combinatorics).

The day ended with Evgeny Vdovin talking about the conjecture that, if a transitive finite permutation group with no soluble normal subgroup has the property that the point stabiliser is soluble, then the group has a base of bounded size (that is, there is a set of points of bounded size whose pointwise stabiliser is the identity). The bound has been variously conjectured to be 7 or 5. This is almost certain to be true. Evgeny’s method shows the importance of choosing the right induction hypothesis. He works down a specially chosen composition series for the group, but the induction hypothesis “there is a base of size *k* cannot be made to work (there are counterexmples). Instead, he has to take the hypothesis “there are at least 5 regular orbits on *k*-tuples”. Why 5? I don’t know, but it seems to work!

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