Existence of nice Folner sequences

I'm attempting a proof by induction and, for the inductive step, it would be very useful for me to have some control on a Folner sequence. Indeed, let $G$ be a finitely generated amenable group, fix a finite symmetric set of generators for $G$ and denote by $B_n$ the ball of radius $n$ in the Cayley graph of $G$ with respect of the fixed generators.

Is it possible to find a Folner sequence $(F_n:n\in\mathbb N)$ for $G$ satisfying the following properties?

(1) $F_1\subseteq F_2\subseteq F_3\subseteq \dots$ and $\bigcup_{n\in\mathbb N}F_n=G$;

(2) for all $n\in \mathbb N$ there exists $k$ such that $B_k\subseteq F_n\subseteq B_{k+1}$.

-
It will work if your group has subexponential growth. – Misha Feb 24 '13 at 0:39
@Misha: well... in that case ${B_n:n\in\mathbb N}$ is a Folner sequence so the interesting case is when you have exponential growth... – Simone Virili Feb 24 '13 at 0:42
@Simone: It may happen that your condition is satisfied only for groups of subexponential growth. I cannot think of a group of exponential growth for which the condition is true. – Mark Sapir Feb 24 '13 at 8:40
@Simone: Folner sets usually have complicated shapes. It was noticed right after the original paper by Folner. I do not remember the references but it (the references) must be in the standard books on amenability (Greenleaf or Wagon). – Mark Sapir Feb 24 '13 at 12:26
What you ask is a bit demanding and might be sensitive on the choice of generating set. On the other hand, Romain Tessera has proved that many groups have such a sequence $(F_n)$ with $B_{cn}\subset F_n\subset B_{Cn}$ for constants $c<C$. These include polycyclic groups and lamplighters over finite groups, see arxiv.org/abs/math/0603138. (The condition $\bigcup F_n=G$ is not hard to require but is usually useless... for instance the Følner sets $[-n,n]$ or $[0,2n]$ in $\mathbf{Z}$ behave the same way.) – YCor Feb 25 '13 at 16:43

This is not possible for $\mathbb{Z}\wr \mathbb{Z}$ and (most) other groups. For the structure of Folner sets there, see, in particular, Erschler, Anna On isoperimetric profiles of finitely generated groups. (English summary) Geom. Dedicata 100 (2003), 157–171. and references there.