In the infinite dihedral group every integer difference |AA^{-1}| - |A^{-1}A| is achievable while in the free group F_2 only even integers are achievable, with constructions and minimum sizes for nonzero difference depending on order-2 elements.
254b, notes 5: Product theorems, pivot arguments, and the larsen--pink non-concentration inequality
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Comparing Left and Right Quotient Sets in Groups
In the infinite dihedral group every integer difference |AA^{-1}| - |A^{-1}A| is achievable while in the free group F_2 only even integers are achievable, with constructions and minimum sizes for nonzero difference depending on order-2 elements.