Hi everyone, Let $\Gamma$ be a finitly generated group. Does someone know of a connection between properties of $\Gamma$ to local symmetries of its Cayley graph? More specificly, what can one learn about $\Gamma$ by looking at the group of isometries of the ball of radius n centered at e (the identity element) in the Cayley graph (reguarding the word length metric)?

I think there is very little that can be said, even when $\Gamma$ is finite. Obviously any automorphism of $\Gamma$ that fixes the connection set of the Cayley graph gives rise to an automorphism of the ball of radius $n$ about $e$; hence if the ball is asymmetric then there is no automorphism of $\Gamma$ that fixes the connection set. (There can be automorphisms of the Cayley graph that do not arise from automorphisms of the group, but I cannot see that these are relevant to your question.)

To get a result of the type you are asking about, you would need to assume that $2n$ is larger than the girth (or your ball would be a "regular" tree, and tell you nothing about the group). But now the balls of radius $n$ will be complex structures, and it is getting difficult to even formulate a result.

The balls do provide information about the spectrum of the adjacency operator, but it is not clear how to relate this to the group structure in general.

In the finite case there is a lot of work devoted to studying automorphism groups of Cayley graphs, but I am not aware of any results of the type you describe.