A compact leaf neither has finite holonomy nor finite fundamental group automatically

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The countable-choice principle used in the foliation pair). The hypotheses "compact leaf", "finite holonomy" and "finite fundamental group" are pairwise distinct for compact leaves.

The boundary torus of the Reeb foliation of the solid torus (The Reeb foliation of the solid torus has the boundary as a leaf, The two-dimensional torus $T^2=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^2$) is compact but has infinite holonomy and infinite fundamental group, and it is not stable, so compactness of the leaf does not imply finiteness of the holonomy group. Conversely a leaf of a product foliation $L\times S^1$ by the slices is compact with trivial holonomy for every closed $L$, including $L=T^2$ with infinite fundamental group, so compactness alone neither implies finite holonomy nor guarantees stability, and "finite holonomy" is strictly weaker than "finite fundamental group" (Finiteness of the fundamental group is sufficient, but not necessary, for Reeb stability).

• The Reeb example. In the Reeb component the holonomy of the loop in the $S^1$-factor is a non-identity contraction germ on the inward half-interval. In the boundaryless glued foliation of $S^3$ (Gluing two Reeb components gives a foliation of the three-sphere), the same torus has a two-sided transversal. Its transport restricts on either Reeb side to the corresponding one-sided transport, so distinct inward iterates give distinct two-sided germs in the exact sense of The holonomy representation and the holonomy group of a leaf. Thus the compact boundary leaf has infinite holonomy; the interior leaves are planes accumulating on it, so it is not stable either. No finiteness of holonomy is available for free.

• The product example. For the product foliation of $L\times S^1$ the leafwise transport in product coordinates is the identity, so every leaf has trivial holonomy regardless of ${\pi }_1(L)$; taking $L=T^2$ exhibits a compact leaf with infinite fundamental group and finite (indeed trivial) holonomy.

• The exact hypothesis. The local Reeb stability theorem of this pair is stated with the hypothesis it actually consumes, finiteness of the holonomy group of the compact leaf; finiteness of ${\pi }_1(L)$ is a sufficient condition for that hypothesis and not a necessary one.