I will start with an argument showing that if \(K\) is a model complete field, then the pure group \(H(K)\) is model complete, where \(H\) is the Heisenberg group (functor), that is \(H(K)=\mathrm{UT}_3(K)\) is the group of 3 by 3 unitriangular matrices with coefficients from \(K\). This is joint work with Maciej Frącek which is based on his Bachelor Thesis.
I will also briefly discuss some other algebraic groups (replacing the Heisenberg group), for example the case of split semisimple groups which is joint work with Daniel Max Hoffmann, Chieu-Minh Tran, and Jinhe Ye.
Time permitting, I will point out some interesting connections with bi-interpretability.
There is nothing wrong with the arguments at this moment in time, but the first version of the Bachelor Thesis problem I gave to Maciej was plain wrong.