Translation lengths in Out(F_n)

TL;DR

This paper proves that all infinite order elements in Out(F_n) have positive translation lengths, leading to new insights into the structure and properties of solvable subgroups within Out(F_n).

Contribution

It establishes a uniform positive lower bound for translation lengths of infinite order elements and derives new structural results for solvable subgroups.

Findings

Infinite order elements in Out(F_n) have positive translation lengths.

Solvable subgroups of Out(F_n) are finitely generated and virtually abelian.

Such subgroups are also quasi-convex.

Abstract

We prove that all elements of infinite order in $Out(F_n)$ have positive translation lengths; moreover, they are bounded away from zero. Consequences include a new proof that solvable subgroups of $Out(F_n)$ are finitely generated and virtually abelian and the new result that such subgroups are quasi-convex.