Some major open problems in group theory can be described within the framework of approximation of groups; for instance, whether all groups are sofic, hyperlinear or MF. We will present the general framework for metric approximation of groups, and particularly with respect to the L^p norm, p \in [1, \infty]. Previous results prove that certain Deligne central extension of lattices in higher rank groups are not L^p approximated for p \in (1, \infty), and we extend this result to the case p=1.

Based on joint work with Alon Dogon and Alex Lubotzky.