The Gap between an automorphism and its Inverse

We reintroduce the function alpha_G (and beta_G) that measures the gap between an (outer) automorphism of $G$ and its inverse. We give an alternative proof of the lower bound for alpha_{F_r} of the free groups, and give an improvement for the lower bound of beta_{F_r}. Furthermore, for the first time, a study of the function alpha_{BS(1,N)} for the Baumslag-Solitar groups BS(1,N), |N|>1, is made, and we prove that it grows linearly. Finally, in an independent way, we define the same concept over the virtual automorphisms and prove that the equivalent function for the free groups has an exponential lower bound