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Yesterday Urban Jezernik and I uploaded to the arxiv our preprint Babai’s conjecture for high-rank classical groups with random generators. I want to use this space to explain what our work is about and where it fits in the literature. To try to cater for a possibly wide range of mathematical backgrounds, I will start friendly and informal and get progressively more precise and technical. Thanks in advance for your attention!

The subject matter is diameters of groups. A familiar example is Rubik’s cube, which has $蚂蚁vp加速器官网$ ( $蚂蚁vp加速器官网$ billion billion) possible configurations. The set of all configurations of the cube can be viewed as a group $旋风vp加速器$ . (For the exact group structure, see wikipedia.) The standard generating set ${S}$ consists of rotations of any of the 6 faces, a set of size $ios 加速器vp$ (including the do-nothing operation). The 旋风vp加速器 is the maximum number of moves it takes to solve the cube from any configuration. Now the number of configurations a distance ${d}$ from the origin, denoted ${S^d}$ , is at most ${19^d}$ : at each step, choose one of the $veee加速器电脑$ possible moves. Therefore the diameter cannot be smaller than

$\displaystyle \left\lceil\frac{\log n}{\log 19}\right\rceil = 16.$

The exact answer is not that much larger: 20. The size of $蚂蚁vp加速器官网$ is plotted below, which exhibits almost perfect exponential growth, until the end when it runs out of space.

In general, let $ios 加速器vp$ be a finite group and $veee加速器电脑$ a set of generators. Assume ${S = S^{-1}}$ and ${1 \in S}$ . The Cayley graph $旋风vp加速器$ is the graph with vertex set ${G}$ and an edge for every pair ${\{g, gs\}}$ with ${g\in G}$ and $ios 加速器vp$ . The diameter ${\mathrm{diam}\,\,\Gamma}$ of a graph ${\Gamma}$ is the “longest shortest path”: the distance between two vertices is the length of the shortest path between them, and the diameter is the greatest distance between any two vertices. The diameter of a Cayley graph ${\mathrm{Cay}(G, S)}$ is the same as the smallest $旋风vp加速器$ such that ${S^d = G}$ .

One expects similar behaviour as we saw for the Rubik’s cube group ${R}$ for any group $旋风vp加速器$ which is sufficiently nonabelian (so that most products of generators are different, causing the balls ${S^d}$ to expand at about the rate they do in a tree). The most ideal form of nonabelianness is simplicity, and Babai’s conjecture is a precise form of this intuition for finite simple groups. The conjecture states that

$\displaystyle \mathrm{diam}\, \mathrm{Cay}(G, S) \leq (\log |G|)^{O(1)},$

uniformly over finite simple groups ${G}$ and generating sets ${S}$ . The trivial lower bound is $ios 加速器vp$ , so the conjecture states that the truth is within a power of this trivial lower bound.

By the classification of finite simple groups, Babai’s conjecture breaks naturally into three parts:

finite groups $ios 加速器vp$ of a fixed Lie type ${X}$ over a field of order ${q}$ with $旋风vp加速器$ ;
quasisimple classical groups of dimension $旋风vp加速器$ with $旋风vp加速器$ and ${q}$ arbitrary: namely, the special linear group ${\mathrm{SL}_n(q)}$ , the symplectic group ${\mathrm{Sp}_n(q)}$ , the derived subgroup of the orthogonal group ${O_n(q)'}$ , and the special unitary group $veee加速器电脑$ ;
alternating groups ${A_n}$ .

The first part is the “bounded rank” case, and the latter two parts make up the “unbounded rank” case (and you can loosely consider $蚂蚁vp加速器官网$ to be ${\mathrm{PSL}_n(1)}$ , if you want).

The bounded rank case of Babai’s conjecture is completely resolved, following breakthrough work of Helfgott, Pyber–Szabo, and veee加速器电脑. Even stronger conjectures can be made in this case. For instance, it may be that the graphs ${\mathrm{Cay}(G, S)}$ have a uniform spectral gap, and in particular that

$\displaystyle \mathrm{diam}\,\,\mathrm{Cay}(G,S) \leq O(\log|G|),$

with the implicit constant depending only on the rank. This is true if the generators are random, by work of Breuillard–Green–Guralnick–Tao.

Personally I am most interested in high-rank groups, such as the alternating group ${A_n}$ (cf. the name of this blog). In the case of ${A_n}$ , Babai’s conjecture (a much older conjecture in this case) asserts simply that

$\displaystyle \mathrm{diam}\, \mathrm{Cay}(A_n, S) = n^{O(1)}.$

For example, if the generating set consists just of the cycles $蚂蚁vp加速器官网$ and $veee加速器电脑$ , then the the diameter is comparable to ${n^2}$ . A folklore conjecture asserts that this is the worst case up to a constant factor. The strongest results in this direction are the result of Helfgott–Seress (see also this later paper of Helfgott) which states that

$ios 加速器vp$

which is quasipolynomial rather than polynomial, and the result of Helfgott–Seress–Zuk which states that, if the generators are random,

$旋风vp加速器$

Thus we are “almost there” with ${A_n}$ . On the other hand, we are still a long way off for high-rank classical groups such as $旋风vp加速器$ . Such a group has size comparable to $veee加速器电脑$ , and Babai’s conjecture asserts that

$\displaystyle \mathrm{diam}\, \mathrm{Cay}(G, S) \leq (n \log q)^{O(1)}.$

In this case the best bound we have is one of Halasi–Maroti–Pyber–Qiao (building on 旋风vp加速器), which states

$旋风vp加速器$

In many ways ${A_n}$ and, say, ${\mathrm{SL}_n(2)}$ behave similarly, but in other ways they are very different. For example, suppose your generating set for ${A_n}$ contains a $veee加速器电脑$ -cycle. Then because there are at most ${n^3}$ ${3}$ -cycles in all, it is clear that every ${3}$ -cycle has length at most $veee加速器电脑$ in your generators, and hence every element of $蚂蚁vp加速器官网$ has length at most ${n^4}$ , so the conjecture is trivial in this case. On other hand, suppose your generating set for ${\mathrm{SL}_n(2)}$ contains the closest analogue to a $ios 加速器vp$ -cycle, a transvection. Unfortunately, there is not a trivial argument in this case, because the number of transvections is roughly ${2^{2n-1}}$ . Indeed this is a difficult problem that was solved only earlier this year by Halasi.

However, if the size of the field ${q}$ is bounded, and you have at least ${q^C}$ random generators, then Urban and I know what to do. Under these conditions we show that

$\displaystyle \mathrm{diam}\, \mathrm{Cay}(G, S) \leq n^{O(1)},$

as conjectured by Babai.

Roughly, the recipe is the following. Let ${S}$ be your set of random generators. We want to show that we can reach everywhere in $ios 加速器vp$ using $旋风vp加速器$ steps in ${S}$ .

Start with a random walk of length $蚂蚁vp加速器官网$ . All of the points you get to will have “large support”, in the sense that ${g-1}$ has no large eigenspace.
Use explicit character bounds and the second moment method to show that, within the first ${n/10}$ steps, you will reach any specified large normal subset $ios 加速器vp$ .
Choose ${\mathfrak{C}}$ so that every element of ${\mathfrak{C}}$ has a ${n^{O(1)}}$ power which has minimal support. Hence we can reach such an element with $旋风vp加速器$ steps.
Now act on that element by conjugation. This looks much like the standard action of ${G}$ on the natural module $旋风vp加速器$ . We show that with high probability the Schreier graph of this action actually has a spectral gap, and in particular logarithmic diameter. Hence we can cover the entire conjugacy class.
Every element of ${G}$ can be written as the product of ${O(n)}$ such elements, so every element of ${G}$ has length ${n^{O(1)}}$ in ${S}$ .

Some of these ideas are recycled from my previous paper with Stefan Virchow (discussed in this earlier post).

Probably the most interesting part is step 4, which generalizes a result of Friedman–Joux–Roichman–Stern–Tillich for the symmetric group. The Schreier graph of the usual action of ${S_n}$ on $旋风vp加速器$ is one of the standard models for a random bounded-degree regular graph, and the spectral gap of this graph is well-studied. The graph we care about on the other hand has vertex set $蚂蚁vp加速器官网$ , with $蚂蚁vp加速器官网$ joined to $ios 加速器vp$ for each ${g}$ in our set of random generators. We found experimentally that, spectrally, this graph is virtually indistinguishable from the random regular graph. In particular, there should be a spectral gap for at least $ios 加速器vp$ random generators.

Our method needs $蚂蚁vp加速器官网$ generators. The reason is that we have great difficulty saying anything about the behaviour of the random walk after about $旋风vp加速器$ steps, or indeed somewhat fewer depending on the type (linear/symplectic/unitary/orthogonal) of the group. To get around this problem we need to ensure that the random walk covers the whole Schreier graph before this critical time, so we need enough generators so that the random walk spreads quickly enough.

I was very sorry to hear today that Jan Saxl died this weekend. Jan was my director of studies at Caius, Cambridge, so was one of my earliest mathematical mentors, and since I eventually moved into group theory he must have been an influential one. Jan could always be relied on for friendly support. I will miss him dearly.

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The College is deeply saddened to hear of the death of long-standing and much-loved Caius Fellow and Professor of Algebra @Cambridge_Uni Professor Jan Saxl on 2 May 2024. pic.twitter.com/lrt0mZOHCC

— Gonville & Caius College (@CaiusCollege) May 4, 2024

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