If a vertex-transitive graph has one ball of polynomially bounded size, it admits a controlled quotient to a Cayley graph of a virtually nilpotent group, with all bounds depending only on the growth ratio.
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A finitary structure theorem for vertex-transitive graphs of polynomial growth
If a vertex-transitive graph has one ball of polynomially bounded size, it admits a controlled quotient to a Cayley graph of a virtually nilpotent group, with all bounds depending only on the growth ratio.