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**Convergence of spherical averages for actions of free groups.**
*(English)*
Zbl 1028.37001

Ergodic theorems (both pointwise and mean) for measure-preserving group actions have a long history, with profound connections to harmonic analysis, probability and amenability. For actions of amenable groups, E. Lindenstrauss [Invent. Math. 146, 259-295 (2001; Zbl 1038.37004)] has recently extended the known results.

In this paper, an associated Markov operator is used to study ‘spherical’ averages for free group actions. Roughly speaking, both pointwise and \(L_1\) ergodic theorems are shown for averages over spheres with even radius (with respect to a fixed set of generators) for functions in \(L\log L\). The limiting function is invariant under the square of the action. The even radius requirement and the invariance only under the square of the action both are needed to deal with the possibility of \(-1\) being an eigenvalue for the action; in the absence of such an eigenvalue averages over spheres converge.

The remarkable results obtained here generalise or strengthen earlier work of R. I. Grigorchuk [Math. Notes 65, 654-657 (1999; Zbl 0957.22006)] and A. Nevo and E. M. Stein [Acta Math. 173, 135-154 (1994; Zbl 0837.22003)]. The paper ends with a discussion of how the results apply to actions of certain Markov groups in the sense of M. Gromov [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)], and a precise conjecture concerning averages over even spheres in Gromov hyperbolic groups.

In this paper, an associated Markov operator is used to study ‘spherical’ averages for free group actions. Roughly speaking, both pointwise and \(L_1\) ergodic theorems are shown for averages over spheres with even radius (with respect to a fixed set of generators) for functions in \(L\log L\). The limiting function is invariant under the square of the action. The even radius requirement and the invariance only under the square of the action both are needed to deal with the possibility of \(-1\) being an eigenvalue for the action; in the absence of such an eigenvalue averages over spheres converge.

The remarkable results obtained here generalise or strengthen earlier work of R. I. Grigorchuk [Math. Notes 65, 654-657 (1999; Zbl 0957.22006)] and A. Nevo and E. M. Stein [Acta Math. 173, 135-154 (1994; Zbl 0837.22003)]. The paper ends with a discussion of how the results apply to actions of certain Markov groups in the sense of M. Gromov [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)], and a precise conjecture concerning averages over even spheres in Gromov hyperbolic groups.

Reviewer: Thomas Ward (Norwich)