2 Clarified definition of i

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?

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# How far can the analogy between a Cayley graph and a symmetric space be pushed?

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 a geodesic symmetry at the identity. 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?