Structural conditions on groups and normal forms guarantee that infinite solutions to quadratic equations have unbounded exponent of periodicity, with the property preserved under graph products.
Eilenberg.Automata, Languages, and Machines, volume A
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Quadratic Equations in Graph Products of Groups and the Exponent of Periodicity
Structural conditions on groups and normal forms guarantee that infinite solutions to quadratic equations have unbounded exponent of periodicity, with the property preserved under graph products.