Let $(M,g)$ be a closed riemannian surface . let $\alpha$ be a simple closed geodesique . does there is exist a simple closed geodesic $\beta$ that intersect alpha at only 1 point p such that $[\alpha]$ and $[\beta]$ does not commute in $\pi\_1(M,p)$ 



intersection of geodesiques 
Yes. Consider the punctured torus, then the $(1, 0)$ and $(0, 1)$ curves together generate the fundamental group (which is the free group on two generators), and so don't commute. Now, if you have a *closed* riemann surface, one of its handles is a punctured torus, so the above construction goes through without change.


**EDIT** The above answer is for hyperbolic surfaces. Obviously, if the fundamental group is abelian (as for the torus or the sphere or the projective plane), the answer is no.


