Uniform difference between the length spectra of Out(F2) and the genus two handlebody group

In this paper, we compare the translation length spectrum of the genus g handlebody group in the Teichmüller space with that of the outer automorphism group of the free group of rank g in the outer space. To this end, we analyze the homomorphism from the handlebody group to the outer automorphism group, induced by the inclusion of the boundary of the handlebody to the handlebody itself. With this homomorphism in hand, we reveal a crucial relationship: the translation length of a fully irreducible outer automorphism provides a lower bound for the translation lengths of pseudo-Anosov maps in its preimage. Notably, in the case of genus two, the difference between the translation length of each fully irreducible outer automorphism and the minimum translation length among the pseudo-Anosov maps in its preimage is bounded above by [Formula: see text]. This result partially addresses a question posed by Hensel and has practical implications, including providing a lower bound for the geodesic counting problem in the genus two handlebody group.