  The entropy, the spectral radius and the drift are important numerical
quantities associated to random walks on countable groups. We prove sharp
inequalities relating those quantities for walks with a finite second moment,
improving upon previous results of Avez, Varopoulos, Carne, Ledrappier. We also
deduce inequalities between these quantities and the volume growth of the
group. Finally, we show that the equality case in our inequality is rather
rigid.
