Every copy of Thompson's group $F$ in $F$ is undistorted

Abstract

Thompson's group $F$ is the group of all piecewise-linear homeomorphisms of the unit interval whose breakpoints are dyadic and whose slopes are integer powers of $2$. If $H$ is a finitely generated subgroup of a finitely generated group $G$, its distortion measures the difference between the intrinsic word metric of $H$ and the metric induced from $G$. The subgroup is undistorted when these metrics are equivalent. Guba and Sapir asked whether $F$ contains a distorted copy of itself. The same question was also suggested by Brin. We answer this question negatively: every subgroup of $F$ isomorphic to $F$ is undistorted in $F$.