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Phase transitions in infinitely generated groups, and related problems in additive number theory

Let A be an infinite set of generators for a group G, and let L_A(r) denote the number of elements of G whose word length with respect to A is exactly r. The purpose of this note is to determine all growth functions L_A(r) associated to infinite generating sets for groups, and to describe a phase transition phenomenon associated with infinite generating sets. A list of open problems is also.included.