**Q1.** Are there "canonical" ways to construct Følner sequences for locally compact amenable groups? Possibly via representations or perhaps characters ?

To elaborate, let $G$ be a l.c. amenable group with the left Haar measure $\lambda$. We say the sequence $F_n$ of compact non-zero measure subsets with $G=\cup A_n$ is a Følner sequence if for any compact set $K$ and $\delta>0$ $$\lambda(KF_n\Delta F_n)\leq \delta \lambda(F_n),$$ for large enough $n$. People sometimes assume extra restrictions such as symmetry of$F_n$ and that each $F_n$ contains of the identity. But let us not worry for about that for now.

**Q2.** Are there ergodic theorems similar to Lindensrauss' for amenable groups with conditions on Følner sequences rather than Shulman's (that is
$\lambda(\cup_{k<n} F^{-1}_kF_n)\leq C\lambda(F_n)$ for all $n$?)

amenable and unimodular. $\endgroup$ – YCor Jan 17 '17 at 8:462more comments