12.88 (1992)

An undirected graph is called a locally finite Cayley graph of a group $G$ if its vertex set can be identified with the set of elements of $G$ in such a way that, for some finite generating set $X = X^{-1}$ of $G$ not containing 1, two vertices $g$ and $h$ are adjacent if and only if $g^{-1}h \in X$. Do there exist two finitely generated groups with the same locally finite Cayley graph, one of which is periodic and the other is not periodic?