TITLE: Examples of normal subgroups of a surface group with no embedded elements
QUESTION [6 upvotes]: 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?

REPLY [8 votes]: 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.