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

Abstract

The final publication is available at Springer via http://dx.doi.org/10.1007/s13348-015-0133-3.For any finitely generated group GG , two complexity functions aGaG and ßGßG are defined to measure the maximal possible gap between the norm of an automorphism (respectively, outer automorphism) of GG and the norm of its inverse. Restricting attention to free groups FrFr , the exact asymptotic behaviour of a2a2 and ß2ß2 is computed. For rank r¿3r¿3 , polynomial lower bounds are provided for arar and ßrßr , and the existence of a polynomial upper bound is proved for ßrßr .Peer ReviewedPostprint (author's final draft

Patterson–Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for outer space

“…Moreover, for any T, S ∈ cv 1 N there exists an (in general non-unique) d L -geodesic path from T to S in cv 1 N , given by natural "folding lines" [24]. The asymmetric distance d L is a useful tool in the study of the geometry of Out(F N ) and it has found significant recent applications, see, for example, [1,2,3,5,24,25,44,53].…”