Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$. There are some obvious candidates to isometries of this graph - for example, translation by elements of $G$, and group automorphisms which preserve $S$.

In some simple cases, these are everything one has (up to composition) - for example, take $G$ be a the free abelian group $G=\mathbb{Z}^{d}$ with $S$ being $\{\pm e_i\}$, with $\{e_i\}$ being the standard basis. This is true for other generating sets as well. However, it is far from correct, say, for the free group with its standard generators.

I am more inclined toward "$\mathbb{Z}^d$-esque" situations in my research, so I would like to try and know more on the first case example.

For the questions themselves:

(a) Consider $G=\mathbb{Z}^d$ with some generating set. Can all isometries of the Cayley graph be seen as a composition of translations and group Automorphisms (preserving S)?

(b) Is this claim true for "nice" nilpotent groups, e.g. the Heisenberg groups?

(c) Must it be false for hyperbolic groups, considering all possible generating sets?

torsion freelattice $\Gamma$ in a simple connected Lie group with finite center and different from $SL(2,\mathbf R)$ has a discrete isometry group by a Theorem of Furman (GAFA 2001). This gives lots of hyperbolic groups, and perhaps the proof can be adapted to show that $\Gamma$ is normal in the isometry group of all its Cayley graphs (which implies that the isometry group of the Cayley is made of group automorphisms). At least for some examples. $\endgroup$1more comment