(Joint with R. Boutonnet, A. Ioana) The notion of local spectral gap for general measure preserving actions will be defined. We prove that the left translation action of a dense subgroup of a simple Lie group has local spectral gap if the subgroup has algebraic entries. This extends to the non-compact setting recent works of Bourgain-Gamburd and Benoist-de Saxce. We also extend Bourgain-Yehudayoff’s result. The application of this result to Banach-Ruziewicz problem, delayed random-walk, and monotone expanders will be explained.