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

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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