Advanced Class Logic

I will talk about some recent work with Tingxiang Zou on higher-dimensional Elekes-Szabó problems in the case of an Ind-constructible action of a group G on a variety X. We expect nilpotent algebraic subgroups N of G to be responsible for any such; this roughly means that if H and A are finite subsets with non-expansion |H*A| <= |A|^{1+\eta}, then H concentrates on a coset of some such N.

A natural example is the action of the Cremona group of birational transformations of the plane. I will talk about a recent result which confirms the above expectation when we restrict to the group of polynomial automorphisms of the plane, using Jung's description of this group as an amalgamated free product, as well as some work in progress which combines weak polynomial Freiman-Ruzsa with effective Mordell-Lang, after Akshat Mudgal, to handle some further special cases.