Finite-dimensional representations constructed from random walks

Abstract

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 $\Vert b(X_n){\Vert }^2/n$ 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 ${\mu }^{\ast n}(e)-{\mu }^{\ast n}(g)$, $n\gg m$, normalized by its average with respect to ${\mu }^{\ast m}$, 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 ${\mu }^{\ast n}-g{\mu }^{\ast n}$ can depend on the ultrafilter $U$ for some groups.