I am interested in the following problem : decide if a certain element of the fundamental group can be represented by a simple closed curve. The general case has already been asked and answered on MO (see [How to detect a simple closed curve from the element in the fundamental group?](https://mathoverflow.net/questions/142587/how-to-detect-a-simple-closed-curve-from-the-element-in-the-fundamental-group)).


I am interested in a more specific case : 


To make things simpler, let us assume we are dealing with a genus $2$ oriented closed surface with its usual set of generator of its fundamental group $a\_1$, $b\_1$, $ a\_2$, $b\_2$. It is clear that the commutator $[a\_1,b\_1]$ can be represented as a simple closed curve : take the curve separating $S$ in two $1$ holed torus in the classic presentation.


My question is the following : is there any other commutator on the free subgroup generated by $a\_1$ and $b\_1$ that can be representated by a simple closed curve ? My attempts to find such a curve all failed, checking on simple ones like $[a\_1, b\_1^2]$, so I wondered whether it was possible or not.



Find a simple closed curve in $S$ which represents a commutator in $\pi_1 S$
The free subgroup generated by $a\_1,b\_1$ is carried by a one-holed torus $T$ embedded in the surface. Let $\gamma$ be a simple closed curve embedded in $T$, and consider $T\smallsetminus \gamma$. A simple argument with Euler characteristic shows that:


* either $T\smallsetminus\gamma$ is the union of a one-hold torus and an annulus, so $\gamma$ was boundary parallel and hence conjugate to $[a\_1,b\_1]$; or
* $T\smallsetminus\gamma$ is a thrice-punctured sphere, so $\gamma$ was non-separating.


In conclusion, every simple closed curve in $\langle a\_1,b\_1\rangle$ is either conjugate to $[a\_1,b\_1]$ or non-separating.


