Kesten's criterion for amenability

Remarks

The left side is the spectral radius of the random walk. It equals the norm of the self-adjoint convolution operator $\lambda(\mu) \colon \xi \mapsto \mu * \xi$ on $\ell^2(G)$, because $\mu^{*2n}(e) = \ip{\lambda(\mu)^{2n} \delta_e}{\delta_e}$. So the criterion is the case of Hulanicki's theorem for this measure: $G$ is amenable exactly when $\norm{\lambda(\mu)} = 1$.

Example: for $G = \Z$ and $\mu = \frac12(\delta_1 + \delta_{-1})$, one has $\mu^{*2n}(0) = \binom{2n}{n} 4^{-n}$, which is about $1/\sqrt{\pi n}$, and its $2n$-th root tends to $1$. For the free group $F_k$ with $k \ge 2$ free generators, and $\mu$ uniform on the generators and their inverses, Kesten computed the spectral radius $\sqrt{2k - 1}/k$, which is less than $1$. So the return probabilities decay exponentially, and $F_k$ is not amenable.

Going deeper

For a student

Kesten's criterion links amenability with probability and with the spectrum of the Laplacian of a Cayley graph. Let $S$ be a finite symmetric generating set and let $\mu$ be uniform on $S$. Then $\lambda(\mu)$ is the Markov operator of the Cayley graph, and amenability is equivalent to $1$ lying in its spectrum. Grigorchuk's cogrowth criterion restates it in terms of the growth of the set of relations. It was used to show that some groups without free subgroups are not amenable.