# Connections between properties of a group and local symmetries of its Cayley graph

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)?

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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.