Abstract: In this talk, we will define the asymptotic h-expansiveness for actions of sofic groups and prove that these give a sufficient condition for the existence of measures of maximal entropy for such actions. We also prove that expansiveness implies asymptotic h-expansiveness and as a consequence it answers a question raised by Lewis Bowen about the existence of measures of maximal entropy for expansive actions. Profinite actions and algebraic actions turn out to be rich sources of examples of asymptotically h-expansiveness being non-expansive. This is joint work with Guo Hua Zhang.