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

TL;DR

This paper explores the geometric and subgroup structure of the automorphism group of Thompson's group F, providing new realizations, presentations, and metric property analyses.

Contribution

It introduces geometric realizations of Aut(F) via periodic tree pair diagrams and analyzes the distortion properties of its subgroups.

Findings

Inner automorphisms are quadratically distorted in Aut(F)

Some subgroups isomorphic to F are undistorted

Effective methods for estimating word length in Aut(F)

Abstract

We describe some of the geometric properties of the automorphism group Aut(F) of Thompson's group F. We give realizations of 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 Aut(F) and their metric properties. In particular, we show that the subgroup of inner automorphisms of F is at least quadratically distorted in Aut(F), whereas other subgroups of Aut(F) isomorphic to F are undistorted.