It seems to be a common statement in textbooks that:
"For a linear wave equation with the dispersion relation $\omega_k$, the propagation speed of information is given by the group velocity."
which is usually demonstrated by the propagation of a wave package centered around a specific $k$ (e.g., wikipedia). But this seems far away from a proof that there is no way to transmit information faster than that.
Generically, assuming there is an upper-bound for the group velocity, is there a rigorous way to prove that it limits the speed of information propagation? Or how should one attack this problem?
Update: thanks to @freecharly, group velocity can be larger than the signal speed according to Milonni's book. I would change the question then: for a generic dispersion relation, how does one decide the maximum speed for information propagation?