  We prove Thompson's group $F$ has quadratic conjugator length function. That
is, for any two conjugate elements of $F$ of length $n$ or less, there exists
an element of $F$ of length $O(n^2)$ that conjugates one to the other.
Moreover, there exist conjugate pairs of elements of $F$ of length at most $n$
such that the shortest conjugator between them has length $\Omega(n^2)$. This
latter statement holds for $T$ and $V$ as well.
