(1) Given a fundamental group representation of a hyperbolic surface, i.e. $<a_j,b_j|\prod[a_j,b_j]=1>$, and given an element in this group, can we determine whether this element can be represented by a simple closed curve?
(2) More specifically, if we consider only a commutative element, whose abelizaions is trivial in the homology group, is there some method to determine whether this element can be represent by a simple closed separating curve on the surface?
For example, $a_1a_2a_1^{-1}a_2^{-1}$ cannot be represented by a simple closed separating curve. I guess the geometric interpretation of a simple closed separating curve is that it is the boundary of the neibourhood of a group of chained curves. Here a group of chained curves means a set of curves $\{c_1, \dots ,c_{2h}\}$ satisfying the following conditions: the geometric intersection numbers are $i(c_j,c_{j+1})=1$ and $i(c_j,c_{k})=0$ for $k-j>1$. But even this criterion is true, the choice of $c_j$'s in the fundamental group are diverse. From an element in the fundamental group, we may not easily see that whether it is the boundary of a group of chained curves. Are there some easy way or algorithm to determine it?
The previous discussion is for question (2). What if we consider the elements in question (1)? I learn from Farb and Margalit's book that there is a neccesary condition on the homology group that the abelization element of a simple closed curve in the homology group must be primitive. What about the representation in the fundamental group?
Best Answer
There is no simple necessary and sufficient condition for whether an element of the fundamental group can be realized by a simple closed curve. However, there are a variety of algorithms known. I believe that the first such algorithm is given in
Reinhart, Bruce L. Algorithms for Jordan curves on compact surfaces. Ann. of Math. (2) 75 1962 209–222.
But probably the most used one is the one in
Birman, Joan S.; Series, Caroline, An algorithm for simple curves on surfaces. J. London Math. Soc. (2) 29 (1984), no. 2, 331–342.
It uses hyperbolic geometry.
I should remark that both of these papers concern unbased curves; however, if $\Sigma_g$ is a closed genus $g$ surface, then an element $\gamma \in \pi_1(\Sigma_g,\ast)$ can be realized by a based simple closed curve if and only if it is freely homotopic to a simple closed curve. Indeed, if $\gamma$ is freely homotopic to a simple closed curve, then there is some $x \in \pi_1(\Sigma_g,\ast)$ such that $x \gamma x^{-1}$ can be realized by a based simple closed curve. But it follows from the Dehn-Nielsen-Baer theorem that inner automorphisms of $\pi_1(\Sigma_g,\ast)$ can be realized by based homeomorphisms of the surface, so thus $\gamma$ can also be realized by a based simple closed curve.