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?
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.