Let $S$ be a closed hyperbolic surface. Suppose $Mod(S)$ denotes the mapping class groups and $T(F)$ denotes the Teichmüller space.
By Birman exact sequence we get the point pushing map $Push:\pi_1(S,p)\rightarrow Mod(S\setminus p)$. By Theorem of Kra we know that $Push(\gamma)$ is pseudo-Anosov if $\gamma$ is a filling curve (i.e. the geometric intersection number of $\gamma$ with every simple closed curve is non-zero).
As $\gamma$ is filling, the length function $l(\gamma)$ on $T(F)$ attains a minimum at the point $Q(\gamma)$ and $Q(\gamma)$ is also unique.
Let $L_{Push(\gamma)}$ be the Teichmüller geodesic fixed by $Push(\gamma)$.
Q) Does $Q(\gamma)$ lies in $L_{Push(\gamma)}$?
