An ergodic theorem for the quasi-regular representation of the free group

Adrien Boyer (1) , Antoine Pinochet Lobos (2)

1 Weizmann Institute of Science [Rehovot, Israël]
2 I2M - Institut de Mathématiques de Marseille

We prove the weak-* convergence of a certain sequence of averages of unitary operators associated to the action of the free group on its Gromov boundary. This result, which can be thought as an ergodic theorem a la von Neumann with coefficients, provides a new proof of the irreducibility of the quasi-regular representation of the free group.

Mots clés: boundary representations; ergodic theorems; irreducibility; equidistribution; free groups

Domaines: Théorie des groupes [math.GR]; Systèmes dynamiques [math.DS]; Théorie des représentations [math.RT]

Adrien Boyer, Antoine Pinochet Lobos. An ergodic theorem for the quasi-regular representation of the free group. Bulletin of the Belgian Mathematical Society - Simon Stevin, 2017, 24 (2), pp.243 - 255. ⟨10.36045/bbms/1503453708⟩. ⟨hal-01587468⟩