# Finite-dimensional representations constructed from random walks

### Anna Erschler

École Normale Supérieure, Paris, France### Narutaka Ozawa

Kyoto University, Japan

## 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 $∥b(X_{n})∥_{2}/n$ tends to a constant in probability, where $X_{n}$ is the trajectory of the random walk $(G,μ)$. As a corollary, we obtain sufficient conditions for $G$ to satisfy Shalom's property $H_{FD}$. Another application is a convergence to a constant in probability of $μ_{∗n}(e)−μ_{∗n}(g)$, $n≫m$, normalized by its average with respect to $μ_{∗m}$, 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 $μ_{∗n}−gμ_{∗n}$ can depend on the ultrafilter $U$ 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