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Let A be an infinite set of generators for a group G, and let LA (r) denote the number of elements of G whose word length with respect to A is exactly r. The growth function LA is a function from the nonnegative integers ℕ0 to the set ℕ0 ∪ {∞}. The purpose of this note is to determine all growth functions associated to infinite generating sets for groups.

Phase Transitions in Infinitely Generated Groups, and Related Problems in Additive Number Theory