A talk about Babai’s PCC conjecture

Tomorrow morning (9am GMT) I am giving a seminar in Novosibirsk (virtually of course). The abstract is below. If you are interested in attending let me know I will give you the Zoom link.

Title: Nonschurian primitive association schemes with many automorphisms

Abstract: This talk is about primitive coherent configurations X of degree n with more than \exp(n^\epsilon) automorphisms, for constant \epsilon > 0. Babai conjectured that all such are so-called Cameron schemes, the orbital configurations of large primitive groups \mathrm{Sym}(m)^{(k)} \wr G for G \le \mathrm{Sym}(d). We will describe several families of examples that show Babai’s conjecture is actually wrong, strictly interpretted. In particular there is a primitive association scheme X of degree n = m^8 for any m \ge 3 such that \mathrm{Aut}(X) is imprimitive and |\mathrm{Aut}(X)| > \exp(m). But a slightly revised form of Babai’s conjecture is still plausible.

Joseph’s conjectures about commuting probability

The commuting probability P(G) of a finite group G is defined to be the probability that two uniformly random elements x, y \in G commute. If G has order n and k conjugacy classes, this is the same thing as k / n. You might think this is a good way of measuring how close a group is to being abelian, but a remarkable and well-known fact is that no finite group satisfies 5/8 < P(G) < 1, so in fact nonabelian groups never get close to being abelian, in this sense.

In the 1970s Keith Joseph made three insightful conjectures about the set of all possible commuting probabilities S = \{P(G) : G~\text{a finite group}\}.

  1. every limit point of S is rational,
  2. for every x \in (0, 1] there is some interval (x-\epsilon, x) that S avoids,
  3. every nonzero limit point of S is in S.

Note that 3 is stronger than 1.

In 2014 I proved the first two of these conjectures (building on an earlier paper of Hegarty), and then I publicly expressed doubt about the third. My doubt was largely based on the observation that there is a family of 2-groups whose commuting probability converges to 1/2, but no 2-group has commuting probability 1/2 (although S_3 does).

But I was wrong! The third conjecture was proved recently in this paper of Thomas Browning:

https://arxiv.org/abs/2201.09402