1. Spectral radius of a random walk
Let be a countable group. Start walking randomly on (for instance, using uniform measure on a symmetric generating set). The Markov operator is
The norm of is called the spectral radius of the random walk,
Kesten’s theorem states that is amenable iff the spectral radius vanishes.
2. Spectral index of a subgroup
Let be an infinite index subgroup of . The spectral index of in is the norm of the Markov operator acting on . One can think of it as a codimension. If is amenable, it equals the r-spectral radius of .
Let act on probability space . One can make spatial averages over . The spectral radius on equals 1, so remove constant functions. The spectral radius on is called the global spectral radius of the random walk. The local spectral radius (on of the orbit of ) is always less,
Definition 1 Say action is Ramanujan if .
Example. Take a tower of finite index subgroups of a free group which does not have property .
Example. Bernoulli actions are Ramanujan. Indeed,