Among the set of all pure states living in a finite dimensional Hilbert space $\mathcal{H}_N$one distinguishes subsets of states satisfying some natural condition. One basis independent choice, consist in selecting the spin coherent states, corresponding to the $SU(2)$ group, or generalized, $SU(K)$ coherent states. Another often studied example is basis dependent, as states coherent with respect to a given basis are distinguished by the fact that the moduli of their off-diagonal elements (called 'coherences') are as large as possible. It is natural to define 'anti-coherent' states, which are maximally distant to the set of coherent states and to quantify the degree of coherence of a given state can by its distance to the set of anti-coherent states. For instance, the separable states of a system composed of two subsystems with $N$ levels are coherent with respect to the composite group $SU(N)\times SU(N)$, while in this setup, the anti-coherent states are maximally entangled.[-]

The study of group actions on Hilbert spaces is central in operator algebras, geometric group theory and representation theory. In many natural situations however, particularily interesting actions on Lp spaces appear for p not 2. One celebrated example is the construction by Pansu (and later greatly generalized by Yu to all Gromov hyperbolic groups) of proper actions of groups of isometries of hyperbolic spaces on Lp for large p. In all these results, the rather clear impression was that it was easier to act on Lp space as p becomes larger. The goal of my talk will be to explain this impression by a theorem and to study how the behaviour of the group actions on Lp spaces depends on p and on the group. In particular, I will show that the set of values of p such that a given countable groups has an isometric action on Lp with unbounded orbits is of the form $[p_c,\infty]$ for some $p_c$, and I will try to compute this critical parameter for lattices in semisimple groups. In passing, we will have to discuss how these objects and properties behave with respect to quantitative measure equivalence. This is a joint work with Amine Marrakchi, partly in arXiv:2001.02490.[-]