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...

According to this survey 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. 

