I have written a monograph on Kähler groups, to be published by Princeton University Press in the spring of 2025, see here to learn more or order it.
Thanks to Benoît Claudon, Sam Hughes, Yuan Liu, Claudio Llosa Isenrich and Mohan Ramachandran for comments on some pre-preliminary versions of this book and to Thomas Delzant for many hours of discussion on the topic of Kähler groups.