I will report on joint works with Adrian Ioana and Alireza Salehi Golsefidi. We study translation actions, that is, actions obtained by left multiplication on a l.c. group G by elements in a countable dense subgroup of G. Such actions preserve the Haar measure, which is infinite when the group G is not compact. In this infinite setting I will explain how to define a notion of local spectral gap which generalizes the usual (very useful) notion of spectral gap from the compact setting.