Revista Matemática Iberoamericana


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Volume 29, Issue 3, 2013, pp. 809–828
DOI: 10.4171/RMI/741

Published online: 2013-08-04

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

José Burillo[1] and Sean Cleary[2]

(1) Universitat Politècnica de Catalunya, Castelldefels, Spain
(2) The City College of CUNY, New York, USA

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

Keywords: Thompson’s group

Burillo José, Cleary Sean: The automorphism group of Thompson’s group $F$: subgroups and metric properties. Rev. Mat. Iberoam. 29 (2013), 809-828. doi: 10.4171/RMI/741