Let $\Sigma$ be a hyperbolic surface possibly with non-empty boundary and punctures.
A closed curve $\gamma$ if filling, if any simple closed curve intersects any representative of free homotopy class of $\gamma$ non-trivially. Typically one uses this definition to (sets) of simple curves, but in this question I am interested in non-simple ones.
I wonder if the following is true:
Question 1. Assume that $\gamma$ is in minimal position, i.e., is immersed, has only double points and no double point is excess. Then $\gamma$ is filling if and only if all the connected components of $\Sigma - \gamma$ are discs, one-punctured discs, or half open annuli.
This would follow from the positive answer to the following:
Question 2: Assume that $\gamma$ is in minimal position. Let $\gamma^g$ be the unique closed geodesic in the free homotopy class of $\gamma$. Is there an isotopy of $\Sigma$, or at least a homeomorphism, which moves $\gamma$ to $\gamma^g$?
Edit: So, it seams that Lemma 2.8 from 'Intersections of curves on surfaces', Hass, Scott, gives a positive answer to Question 1 for all orientable surfaces.
The answer for Question 2 as it is stated is possibly no. Although the answer is yes, if one allows to do some modifications to $\gamma$. Namely, by Hass and Scotts disk flow (see e.g. 'A Whitehead algorithm for surface groups', Levitt, Vogtmann) we can perform a series of 'push a part of $\gamma$ across a double point' moves (Rademaister like moves) on $\gamma$ to obtain $\gamma_1$, such that there is an isotopy of $\Sigma$ moving $\gamma_1$ to $\gamma^g$.
I don't delete this question since maybe someone might be interested.
Best Michal
