Profinite rigidity amongst Kähler groups

I will report on joint work with Sam Hughes, Caludio Llosa Isenrich, Pierre Py, Ryan Spitler, and Stefano Vidussi that proves, among other things, profinite rigidity of fundamental groups of products of Riemann surfaces among fundamental groups of compact Kähler manifolds. Essential to the proof is profinite rigidity of the collection of homomorphisms to cocompact Fuchsian groups.