groups – Page 3 – Random Permutations

I just uploaded to the arxiv my paper Diameter of classical groups generated by transvections. This paper is a small further step in the direction of Babai’s conjecture on the diameter of simple groups, which predicts for any (nonabelian) finite simple group ${G}$ and for any generating set ${S}$ , the number of elements of ${S}$ needed to represent any element of ${G}$ is bounded by ${(\log |G|)^C}$ for some constant ${C}$ . Roughly this means that finite simple groups are “well-connected”, and starting from any generators you can “find” any element of the group rapidly.

The “Pyber programme” suggests that Babai’s conjecture is really three problems. Assume ${G}$ is either ${A_n}$ or a classical group with defining module ${\mathbf F_q^n}$ , say ${\mathrm{SL}_n(q)}$ . The degree of an element ${g \in G}$ is the size of its support if ${G = A_n}$ or the rank of ${g - 1}$ if ${G}$ is classical. Now the three problems are:

Given arbitrary generators, find an element of degree at most ${0.9 n}$ .
Given a set of generators including an element of degree at most ${0.9 n}$ , find an element of minimal degree.
Given a set of generators including an element of minimal degree, find everything.

(In each problem, “find” really means “establish the existence of an element whose length in the generators is bounded by ${(\log |G|)^C}$ ”.)

The third problem is actually trivial for ${A_n}$ . Note that ${\log |A_n| = \log n! \approx n \log n}$ , so “polynomial in ${\log |G|}$ ” = “polynomial in ${n}$ ”. Now suppose a set of generators includes a ${3}$ -cycle (an element of minimal degree in the alternating group). There are fewer than ${n^3}$ such ${3}$ -cycles, and ${G}$ acts on the ${3}$ -cycles transitively by conjugation, so all ${3}$ -cycles have length at most ${n^3}$ in the generators. Every element of ${A_n}$ can be written as the product of at most ${n}$ such ${3}$ -cycles, so every element of ${A_n}$ has length at most ${n^4}$ in the generators.

But for classical groups the third problem is highly nontrivial. In the case of ${G = \mathrm{SL}_n(q)}$ , the elements of minimal degree are the transvections. The number of transvections is comparable to ${q^{2n}}$ , i.e., exponential in ${n}$ , so such a cheap argument is not available. But, the third problem is now solved anyway (with the exception of orthogonal groups — see below!). This is what my paper is about. The case of odd characteristic had been done already by Garonzi, Halasi, and Somlai (omitting some cases in characteristic ${3}$ ), but the (most complicated) case of characteristic ${2}$ is covered in my paper.

The basic approach, following the previous work of Garonzi, Halasi, and Somlai, is to reduce as much of the algebra as possible to combinatorics. We can make an analogy with the symmetric group ${S_n}$ . Let ${T \subset S_n}$ be a set of transpositions ${(ij)}$ (where ${1 \le i < j \le n}$ ). The transposition graph ${\Gamma}$ is defined to have vertex set ${\{1, \dots, n\}}$ and an edge ${i \sim j}$ whenever ${(ij) \in T}$ . Let ${G}$ be the group generated by ${T}$ . Then the following are equivalent:

${\Gamma}$ is connected,
${G}$ is transitive,
${G = S_n}$ .

This illustrates one simple way in which algebra can be reduced to combinatorics. There is an analogous object for transvections called the transvection graph (though really it is a directed graph), and it can be used to understand when a set of transvections in ${\mathrm{SL}_n(q)}$ generates an irreducible subgroup, but it is necessary to introduce several more gadgets (mostly combinatorial) to understand whether they actually generate ${\mathrm{SL}_n(q)}$ .

Most of the work in the paper involves leveraging a theorem of Kantor classifying the irreducible subgroups generated by transvections. For each type of such subgroup, we try to build a combinatorial “certificate” that our set of transvections is not trabbed in that type of subgroup. For example, if our transvections are not trapped in a symplectic group then there must be a “nonsymplectic cycle” in the transvection graph. This industrial use of Kantor’s theorem is the novel idea of my paper.

What about orthogonal groups? What I really mean is ${\Omega_n^\varepsilon(q)}$ , the index-2 subgroup of ${\mathrm{SO}_n(q)}$ . These groups do not contains transvections! The elements of minimal degree have degree ${2}$ ; they are called “long root elements”. Kantor’s theorem also classifies subgroups generated by long root elements, so in principle this problem can be approached in broadly the same way, but we need a suitable variant of the transvection graph (the “long root element graph”?). Even more generally, it does not seem out of reach to prove Babai’s conjecture under the hypothesis that the generators contain an element of bounded degree.

Do these problems sound interesting to you? Are you looking for PhD opportunities in algebra or combinatorics? Consider coming to Belfast to do your PhD! Write to me by email if you are interested (with an introduction and CV).

On a personal note, this paper was quite a slog, and slowly came together over the course of about 8 months of off-and-on effort. That time is not proportionally reflected in the length of the paper but it may be in the technicality of some of the arguments. A particularly challenging part was coming up with a certificate (as described above) for symmetric-type subgroups, i.e., irreducible subgroups of ${\mathrm{SL}_n(q)}$ isomorphic to either ${S_{n+1}}$ or ${S_{n+2}}$ (acting on the deleted permutation module). In the end the argument is not especially difficult but it took a lot of care getting everything right.

By contrast, another paper I recently put on the arxiv was the result of a moment of inspiration and only a day or so of working out the details and writing up. Which paper will get into a better journal I wonder?

1. Boston–Shalev for conjugacy classes

Last week Daniele Garzoni and I uploaded to the arxiv a preprint on the Boston–Shalev conjecture for the conjugacy class weighting. The Boston–Shalev conjecture in its original form predicts that, in any finite simple group $G$ , in any transitive action, the proportion of elements acting as derangements is at least some universal constant $c > 0$ . This conjecture was proved by Fulman and Guralnick in a long series of papers. Daniele and I looked at conjugacy classes instead, and we found an analogous result to be true: the proportion of conjugacy classes containing derangements is at least some universal constant $c' > 0$ .

Our proof depends on the correspondence between semisimple conjugacy classes in a group of Lie type and polynomials over a finite field possibly with certain restrictions: either symmetry or conjugate-symmetry. We studied these sets of polynomials from an “anatomical” perspective, and we needed to prove several nontrivial estimates, e.g., for

the number of polynomials with a factor of a given degree (which is closely related the “multiplication table problem”),
the number of polynomials with an even or odd number of irreducible factors,
the number of polynomials with no factors of small degree,
or the number of polynomials factorizing in a certain way (e.g., as $f = gg^*$ , $g$ irreducible, $g^*$ the reciprocal polynomial).

For a particularly neat example, we found that, if the order of the ground field is odd, exactly half the self-reciprocal polynomials have an even number of irreducible factors — is there a simple proof of this fact?

2. Growth in Linear Groups

Yesterday Brendan Murphy, Endre Szabo, Laci Pyber, and I uploaded a substantial update to our preprint Growth in Linear Groups, in which we prove one general form of the “Helfgott–Lindenstrauss conjecture”. This conjecture asserts that if a symmetric subset $A$ of a general linear group $\mathrm{GL}_n(F)$ ( $n$ bounded, $F$ an arbitrary field) exhibits bounded tripling, $|A^3| \le K|A|$ , then $A$ suffers a precise structure: there are subgroup $H \trianglelefteq \Gamma \le \langle A \rangle$ such that $\Gamma / H$ is nilpotent of class at most $n-1$ , $H$ is contained in a bounded power $A^{O_n(1)}$ , and $A$ is covered by $K^{O_n(1)}$ cosets of $\Gamma$ . Following prodding by the referee and others, we put a lot more work in and proved one additional property: $\Gamma$ can be taken to be normal in $\langle A \rangle$ . This seemingly technical additional point is actually very subtle, and I strongly doubted whether it was true late into the project, more-or-less until we actually proved it.

We also added another significant “application”. This is not exactly an application of the result, but rather of the same toolkit. We showed that if $G \le \mathrm{GL}_n(F)$ (again $F$ an arbitrary field) is any finite subgroup which is $K(n)$ -quasirandom, for some quantity $K(n)$ depending only on $n$ , then the diameter of any Cayley graph of $G$ is polylogarithmic in the order of $|G|$ (that is, Babai’s conjecture holds for $G$ ). This was previously known for $G$ simple (Breuillard–Green–Tao, Pyber–Szabo, 2010). Our result establishes that it is only necessary that $G$ is sufficiently quasirandom. (There is a strong trend in asymptotic group theory of weakening results requiring simplicity to only requiring quasirandomness.)

The intention of our paper is more-or-less to “polish off” the theory of growth in bounded rank. By contrast, growth in high-rank simple groups is still poorly understood.

3. The Kelley–Meka result

Not my own work, but it cannot go unmentioned. There was a spectacular breakthrough in additive combinatorics last week. Kelley and Meka proved a Behrend-like upper bound for the density of a subset $A \subset \{1, \dots, n\}$ free of three-term arithmetic progressions (Roth’s theorem): the density of $A$ is bounded by $\exp(-c (\log n)^\beta)$ for some constants $c, \beta > 0$ . Already there are other expositions of the method which are also worth looking at: see the notes by Bloom and Sisask and Green (to appear, possibly).

Until this work, density $1 /\log n$ was the “logarithmic barrier”, only very recently and barely overcome by Bloom and Sisask. Now that the logarithmic barrier has been completely smashed, it seems inevitable that the new barometer for progress on Roth’s theorem is the exponent $\beta$ . Kelley and Meka obtain $\beta = 1/11$ , while the Behrend construction shows $\beta \le 1/2$ .