Finiteness of the fundamental group is sufficient, but not necessary, for Reeb stability

(Sufficiency.) Suppose ${\pi }_1(L,x)$ is finite. Then $\mathrm{Hol}(L,x)={\rho }_x({\pi }_1(L,x))$ is the image of a finite group under the homomorphism ${\rho }_x$, hence finite [F1, F2]. By local Reeb stability the compact leaf with finite holonomy is stable [F3]. Thus finiteness of the fundamental group is sufficient for stability of $L$.

(Non-necessity.) Consider the product foliation of $L\times {\mathbb{R}}^q$ by the slices $L\times \{t\}$ for a compact leaf $L$. Every leaf is compact and diffeomorphic to $L$, and the leafwise transport of a transversal $\{y\}\times {\mathbb{R}}^q$ in product coordinates is the identity, so the holonomy representation is trivial and the leaf has a fundamental system of product neighbourhoods [F4]. This applies in particular when ${\pi }_1(L,x)$ is infinite, so finiteness of the fundamental group is not necessary for stability; and since trivial holonomy is finite, "finite holonomy" is strictly weaker than "finite fundamental group" for compact leaves.

Therefore on compact leaves the hypothesis consumed by local Reeb stability is finiteness of the holonomy group; finiteness of ${\pi }_1$ implies it and is sufficient, while product foliations with infinite ${\pi }_1$ show it is not necessary.