We show that every externally definable fsg group in an NIP structure M is definably isomorphic to a group interpretable in M. Our proof relies on honest definitions and a measure theoretic group chunk theorem reconstructing a hyper-definable group from a generically given group operation. We also discuss some preliminary results and directions on externally interpretable fsg groups, and on going beyond fsg.

The Grothendieck ring of a first order structure was introduced by Krajìček-Scanlon and Denef-Loeser, and is the universal ring classifying definable sets up to definable bijections. Alternatively, one may view this ring as a universal Euler characteristic on definable sets. I will give an introduction to these Grothendieck rings and give several examples. Afterwards I will focus on valued fields, and discuss an Ax-Kochen/Ershov principle for computing the Grothendieck ring in terms of the residue field and value group. Such an approach was introduced by Hrushovski-Kazhdan in the algebraically closed case, and we extend it to more general henselian valued fields. This is based on joint work with Mathias Stout.