Finite-dimensional representations constructed from random walks
Anna Gennad'evna Erschler, Narutaka Ozawa · Commentarii Mathematici Helvetici · 2018
Given a 1-cocycle b with coefficients in an orthogonal representation, we show that every finite dimensional summand of b is cohomologically trivial if and only if \| b(X_n) \|^2/n tends to a constant in probability, where X_n is the trajectory of the random walk (G,\mu) . As a corollary, we obtain sufficient conditions for G to satisfy Shalom's property H_{\mathrm{FD}} . Another application is a convergence to a constant in probability of \mu^{*n}(e) -\mu^{*n}(g) , n\gg m , normalized by its average with respect to \mu^{*m} , for any finitely generated infinite amenable group without infinite virtually abelian quotients. Finally, we show that the harmonic equivariant mapping of G to a Hilbert space obtained as an U -ultralimit of normalized \mu^{*n}- g \mu^{*n} can depend on the ultrafilter U for some groups.