Partial Dehn twists of free groups relative to local Dehn twists - a dichotomy

This paper is part of my PhD thesis, and I would like to thank sincerely my advisors, Arnaud Hilion and Martin Lustig, for their advice and encouragement.A criterion for quadratic or higher growth of group automorphisms is established which are represented by graph-of-groups automorphisms with certain well specified properties. As a consequence, it is derived (using results of a previous paper of the author) that every partial Dehn twist automorphism of $\FN$ relative to local Dehn twist automorphisms is either an honest Dehn twist automorphism, or else has quadratic growth