Is it true that the fundamental groups of compact Kahler surfaces have property T if and only if it they are finite? I am having trouble finding counterexamples to this, but maybe that's just me...



Kazhdan's property T for Kahler surfaces
According to this [survey](http://library.msri.org/books/Book28/files/arapura.pdf) by Donu Arapura, Toledo proved that many arithmetic lattices in higher rank algebraic $\mathbb{Q}$-groups (with hermitian symmetric space) are fundamental groups of smooth *projective* surfaces.


In particular $Sp(2n,\mathbb{Z})$ for $n>2$, is such a group, and has property (T).


Note that once you get a group as fundamental group of a smooth projective variety you obtain a smooth projective surface with the same fundamental group by intersecting with some generic hyperplanes.


