Finite-dimensional representations constructed from random walks

  • Anna Erschler

    École Normale Supérieure, Paris, France
  • Narutaka Ozawa

    Kyoto University, Japan


Given a 1-cocycle with coefficients in an orthogonal representation, we show that every finite dimensional summand of is cohomologically trivial if and only if tends to a constant in probability, where is the trajectory of the random walk . As a corollary, we obtain sufficient conditions for to satisfy Shalom's property . Another application is a convergence to a constant in probability of , , normalized by its average with respect to , for any finitely generated infinite amenable group without infinite virtually abelian quotients. Finally, we show that the harmonic equivariant mapping of to a Hilbert space obtained as an -ultralimit of normalized can depend on the ultrafilter for some groups.

Cite this article

Anna Erschler, Narutaka Ozawa, Finite-dimensional representations constructed from random walks. Comment. Math. Helv. 93 (2018), no. 3, pp. 555–586

DOI 10.4171/CMH/444