Limit cycles of a leaf

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $F$ be a transversely oriented codimension-one foliation, $L$ a leaf, and $x\in L$. For a chosen side $j$ of $L$, let $N_j(L,x)◃{\pi }_1(L,x)$ be the subgroup of classes whose one-sided normal- fence holonomy germ is the identity. The quotient $P_j(L,x)={\pi }_1(L,x)\mathrm{/}{N}_j(L,x)$ is Novikov’s one-sided limit-cycle group. A class $[\alpha ]$ is a limit cycle on side $j$ precisely when its image in $P_j(L,x)$ is nonidentity, equivalently its one-sided holonomy germ is nonidentity. The two groups $P_{+}(L,x)$ and $P_{-}(L,x)$ record ordinary right and left limit cycles.

The separate limitwise-nullhomotopy subgroup ${\mathrm{\Pi }}_1^j(L,x)$ is defined later on this page as a set of classes in $N_j(L,x)$ whose sufficiently small displaced loops are nullhomotopic in their leaves. Its containment in $N_j(L,x)$ is part of that definition; it is distinct from the ordinary limit-cycle quotient $P_j(L,x)$ defined here.