As a consequence of the loop theorem, if $F$ is a closed surface in the boundary of a 3-manifold and if the kernel $N = \ker(\pi\_1(F) \to \pi\_1(M))$ is nonempty then there is a nontrivial element of $N$ that can be represented by an embedded curve. I read in Hempel's book that there are lots of normal subgroups of $\pi\_1(F)$ that contain no nontrivial elements that can be represented by embedded curves. How can I cook up such subgroups and prove that they have this property? 



Examples of normal subgroups of a surface group with no embedded elements
One source of interesting examples is the lower central series. In my paper


J. Malestein, A. Putman,
On the self-intersections of curves deep in the lower central series of a surface group,
Geom. Dedicata 149 (2010), no. 1, 73–84.


my coauthor and I show that the minimal number of self-intersections among nontrivial elements of the kth term of the lower central series of a surface group goes to infinity as k goes to infinity.


