## Notes of Miklos Abert’s lecture

1. Spectral radius of a random walk

Let ${G}$ be a countable group. Start walking randomly on ${G}$ (for instance, using uniform measure on a symmetric generating set). The Markov operator is

$\displaystyle \begin{array}{rcl} M:\ell^2G\rightarrow\ell^2G,\quad (fM)(x)=\frac{1}{|S|}\sum_{s\in S}f(xs). \end{array}$

The norm of ${M}$ is called the spectral radius of the random walk,

Kesten’s theorem states that ${G}$ is amenable iff the spectral radius vanishes.

2. Spectral index of a subgroup

Let ${H}$ be an infinite index subgroup of ${G}$. The spectral index of ${H}$ in ${G}$ is the norm of the Markov operator acting on ${\ell^2(Schreier(G/H,S))}$. One can think of it as a codimension. If ${H}$ is amenable, it equals the r-spectral radius of ${G}$.

Let ${G}$ act on probability space ${(X,\mu)}$. One can make spatial averages over ${(X,\mu)}$. The spectral radius on ${L^2(X,\mu)}$ equals 1, so remove constant functions. The spectral radius on ${L_0^2(X,\mu)}$ is called the global spectral radius ${\rho_{glob}}$ of the random walk. The local spectral radius ${\rho_{loc}}$ (on ${\ell^2}$ of the orbit of ${x}$) is always less,

$\displaystyle \begin{array}{rcl} \rho_{loc}(x)\leq\rho_{glob}. \end{array}$

Definition 1 Say action is Ramanujan if ${\rho_{loc}(x)=\rho_{glob}}$.

Example. Take a tower of finite index subgroups of a free group which does not have property ${\tau}$.

Example. Bernoulli actions are Ramanujan. Indeed,

Theorem 2