The automorphism group of Thompson’s group $F$: subgroups and metric properties

Abstract

We describe some of the geometric properties of the automorphism group $\mathrm{Aut}(F)$ of Thompson’s group $F$. We give realizations of $\mathrm{Aut}(F)$ geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of $\mathrm{Aut}(F)$ and their metric properties. In particular, we show that the subgroup of inner automorphisms of $F$ is at least quadratically distorted in $\mathrm{Aut}(F)$, whereas other subgroups of $\mathrm{Aut}(F)$ isomorphic to $F$ are undistorted.