Summary: Translation lengths in Out(F n )
We prove that all elements of infinite order in Out(F n ) have posi≠
tive translation lengths; moreover, they are bounded away from zero.
As a consequence we get a new proof that solvable subgroups of
Out(F n ) are finitely generated and virtually abelian.
In this paper we will study the translation lengths of outer automorphisms
of a free group. Following [GS91] we define the translation length Ý X;G (g) of
g 2 \Gamma to be
kg n k
where \Gamma is a group with finite generating set X, and kgk denotes the length
of g in the word metric on \Gamma associated to X.
Farb, Lubotzky and Minsky proved that Dehn twists (more generally,
all elements of infinite order) in Mod(\Sigma g ) have positive translation length
([FLM]). We prove