Convergence of spherical averages for actions of free groups
  Convergence of non-uniform spherical averages is obtained for measure-preserving actions of free groups. This result generalizes theorems of Grigorichuk, Nevo and Stein [in particular, a simpler proof of the Nevo-Stein theorem about uniform spherical averages is obtained.]   The proof uses the Markov operator approach, first proposed by R.I. Grigorchuk. To a measure-preserving action of a free group and a matrix of weights, a Markov operator is assigned in such a way that convergence of spherical averages with corresponding weights is equivalent to convergence of powers of the Markov operator. That last is obtained using Rota's "Alternierende Verfahren"; the 0-2 law for Markov operators in the form of Kaimanovich; and a suitable maximal inequality. 
Alexander I. Bufetov