In the first talk, I will recall the definition of acylindrical hyperbolicity and state a standard criterion to prove the acylindrical hyperbolicity of a group by means of an appropriate strongly contracting element. I will also explain why, except for the case of actions on trees, applying this criterion is generally cumbersome for actions on non locally compact spaces.

In the second talk, I will introduce a strengthening of the notion of strongly contracting element which is particularly suitable in the case of actions on non locally compact spaces. We will use this to obtain a more manageable criterion and prove (or reprove) the acylindrical hyperbolicity of many groups acting on CAT(0) cube complexes: Many right-angled Artin groups, the Higman group, some subgroup of the Cremona group Bir(P^3(C)).

This is joint work with I. Chatterji.