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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 $G$...

A well-known property on groups acting on trees is: Theorem: Let $T$ be a tree and $g,h \in \mathrm{Isom}(T)$ two elliptic isometries. If $\mathrm{Fix}(g) \cap \mathrm{Fix}(h) = \emptyset$ then the ...