I was quite intrigued by Tutte polynomials in a recent talk I had been to. It was introduced as a polynomial associated to a undirected finite graph. For a graph $G=(V,E)$ we form the polynomial

$T_G(x,y)=\sum_{A\subseteq E} (x-1)^{k(A)-k(E)}(y-1)^{k(A)+|A|-|V|}$

where $k(A)$ is the number of components of the graph $(G,A)$. 

I have heard and read about the significance of this polynomial. However, I'm completely new to combinatorics and graph theory and have no prior knowledge of what is or isn't known about these things. I have two questions which should be easy enough for the experts.

(1) Let us start with a finite group $G$, chose a presentation and consider its Cayley graph $\Gamma$. Now consider the Tutte polynomial $T_\Gamma$ of $\Gamma$. We suppose that $G$ is the  fundamental group of some *reasonable* space $M$ (it always is $\pi_1$ of some space but what I mean here is the space is a manifold or very broadly an object where distances and volumes make sense). 

**Question** Does the Tutte polynomial $T_\Gamma$ encode any information about the topological invariants of $M$?
 
(2) In my own naive way I was trying to see if anything similar (like the formula at the outset) works for infinite graphs. The existence of such a thing, at least, looks very interesting as Tutte polynomials generally have a lot of significance. If such a thing doesn't exist, one may begin to study properties of the Tutte polynomials associated to finite subgraphs of an infinite graph and analyze the asymptotics of the these properties on the poset of finite subgraphs. 

**Question** Is there a known way to define Tutte polynomials for infinite graphs? If so, how is defined and what is known? 

**Question** If the above answer is yes then what does $T_\Gamma$ encode about the growth of $G$ when $G$ is the fundamental group of a manifold? Note that by the $\check{S}varc$-$Milnor$ lemma this growth is asymptotic to the growth (in the sense of geometry) of the universal cover of $M$.  

__________________________________________________________________

REMARK : I edited the first question as the growth of a finite group is not terribly interesting. The original first question should make more sense now, as presented, as the last question.