Finite-Dimensional Representations constructed from Random Walks

Abstract

Given a $1$-cocycle $b$ with coefficients in an orthogonal representation, we show that any 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 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.

22 pages; typos fixed (v2); Section 4 revised to cover locally compact groups (v3)

References in corpus (1)

• A functional analysis proof of Gromov's polynomial growth theorem

Cited by in corpus (2)

• Shalom's property $H_{\mathrm{FD}}$ and extensions by $\mathbb{Z}$ of locally finite groups
• Harmonic cocycles, von Neumann algebras, and irreducible affine isometric actions