**Updates:** Changed a bit the definition to include infinite dimensional Banach spaces; Included `questions 0`

and `subquestion`

. Improved (I hope) the presentation.

The growth rate of a group, being a metric property, should have some reformulation just in terms of metric spaces. Also the amenability, thinking of Folner's characterization (which basically means that we can find arbitrarily big set with very small boundary), should have some reformulation in terms of metric space (there are indeed some). Now, before proving that convex-like structures in the sense of Nate Brown are embeddable into Banach spaces and solving (even too easily) all my problems, I have spent some time playing with some strange metric spaces, catching a general property that seems to do *some* job.

Let $(X,d)$ be a metric space. If $x\in X$ and $R>0$, I denote with $B(x,R)$ the open ball and with $S(x,R)$ the sphere around $x$ with radius $R$.

Definition:Let $p_R(x)$ be the number (possibly infinite) of open, continuous and injective maps $f:B(x,R)\rightarrow X$ such that

- $f(B(x,R))\subseteq S(x,R)$
- the images $f(B(x,R))$ are mutually disjoint
I say that $(X,d)$ has the property SB (Small Boundary) if for all $x\in X$ one has $\sum_{R>0}p_R(x)<\infty$.

Examples: Banach spaces have this property. The metric linear space $(\mathbb R,d)$, with $d(x,y)=min(1,|x-y|)$ does not have SB (indeed $p_1(0)=\infty$).

Now let $G$ be a finitely generated group with a fixed symmetric set of generators and consider the word distance with respect to this generating set. Let $x=e$, being $e$ the identity element in $G$. Besides the questions number 0: `does $p_R(e)$ depend on $S$? (maybe)`

; and `does the $\sup$ of the definition depend on $S$? (.... sometimes I believe in God ...)`

I guess that property SB should have some relation with the amenability and with the growh rate of $G$. For instance it seems to me quite evident that Subexponential growth implies SB. A very natural question now would be

Question 1:Does amenability imply SB?

It seems to me that one property that should be true for groups and that would make some thing easier is

Subquestion:Is $p_R(e)$ non-decreasing in $R$?

another interesting question would be *so,e* converse of question 1:

Question 2:Does SB for metric spaces (probably with some extra-condition) implies the existence of somegoodmeasure?

I have no more ideas at the moment, but still think that would be nice to find some (maybe not this one) unifying concept.

Thanks in advance for any comments,

Valerio1.