Bounding the gap between a free group (outer) automorphism and its inverse

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For any finitely generated group G, two complexity functions αG and βG are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of G and the norm of its inverse. Restricting attention to free groups Fᵣ, the exact asymptotic behaviour of α₂ and β₂ is computed. For rank r 3, polynomial lower bounds are provided for αᵣ and βᵣ, and the existence of a polynomial upper bound is proved for βᵣ.