# Groups and Geometries 2019

I just got back from a conference at Banff International Research Station on “Groups and Geometries”. Many thanks to Martin Liebeck, Inna Capdebosq and Bernhard Muehlherr for organising a brilliant week.

I gave a talk at the conference entitled “The relational complexity of a finite primitive permutation group”, which you can view by clicking here. The abstract of the talk was the following:

Motivated by questions in model theory, Greg Cherlin introduced the idea of “relational complexity”, a statistic connected to finite permutation groups. He also stated a conjecture classifying those permutation groups with minimal relational complexity. We report on recent progress towards a proof of this conjecture. We also make some remarks about permutation groups with large relational complexity, and we explain how this statistic relates to others in the literature, notably base-size.

This work is joint with Pablo Spiga, Martin Liebeck, Francesca Dalla Volta, Francis Hunt and Bianca Lodà.”

**Erratum**: at minute 39 of the talk I mis-stated a theorem. The correct statement is as follows:

**Theorem** (Gill, Lodà, Spiga) There exists a constant $c$ such that if $G$ acts primitively on a set $X$ of size $t$, then either

- $H(G,X) < c {\rm log} t$
- $G$ is a subgroup of $S_m \wr S_r$ containing $(A_m)^r$, where the action of $S_m$ is on $k$-element subsets of ${1, …, m}$ and the wreath product has the product action of degree $t={m \choose k}^r$.