If $G$ is a finitely group and $S$ a finite symmetric set of generators, the associated Cayley graph, then $i: x \mapsto x^{-1}$ is gives rise to a geodesic symmetry $i$ at the identity:
If $g=s_1^{e_1}\cdots s_k^{e_k}$ with $e_i \in \{\pm 1\}$, then let $i(g):=s_1^{-e_1}\cdots s_k^{-e_k}$.
Translating $i$ via $G$, there is a symmetry at every vertex.
For points in the interior of edges, there is likewise a (local) symmetry.
This prompts my question: What concepts of locally symmetric Riemannian spaces can be applied to the study of Cayley graphs?
