A compact C¹ leaf has finitely generated fundamental group

(A $C^1$ defining function.) By [F1] the leaf $L$ is a compact embedded $C^1$ hypersurface; cover $L$ by finitely many foliation charts whose transverse coordinate $t_i$ vanishes exactly on $L$ and is positive on the cooriented positive side, and let ${\chi }_i$ be a smooth partition of unity subordinate to these charts [F1, F2]. The weighted sum $f:={\sum }_i{\chi }_i{t}_i$, extended by zero outside the supports, is a $C^1$ function on a neighbourhood of $L$; it vanishes on $L$, and at every $p\in L$ its differential $d{f}_p={\sum }_i{\chi }_i(p)d(t_i{)}_p$ is a positive multiple of the coorientation conormal, because every active $d(t_i{)}_p$ is such a positive multiple [F1, F2]. In particular $d{f}_p\mathrm{\ne }0$ on $L$.

(Smooth defining function and flow collar.) Since $L$ is compact and $df\mathrm{\ne }0$ along it, a finite subcover argument and a partition of unity produce a smooth vector field $V$ and a constant $c>0$ with $df(V)>c$ on a neighbourhood of $L$ [F2]. The flow $\mathrm{\Phi }$ of $V$ exists there for a uniform time by compactness, and its derivative at $(p,0)$ is invertible, so by the inverse function theorem the flow is locally a collar of $L$. It is globally injective after shortening the time interval: otherwise, from pairs with equal image and times tending to zero, compactness gives a subsequence converging to two points of $L$ with equal image, hence to the same point; both pairs then lie in a single local inverse neighborhood, a contradiction. Thus it gives a collar in which $f$ is strictly increasing along the flow lines, one on each side of $L$ [F4]. Mollify $f$ on a compact subcollar by a finite-chart mollifier argument: decompose $f$ with a finite smooth partition, extend each compactly supported chart expression by zero, convolve with a standard mollifier, and use uniform continuity of $f$ and its first derivatives on the compact supports to obtain a smooth function $\tilde{f}$, arbitrarily $C^1$-close to $f$ [F3]. Choose $\tilde{f}$ close enough that $d\tilde{f}(V)>c\mathrm{/}2$ and that its values at the two ends of every flow segment have opposite signs; then $d\tilde{f}$ is nowhere zero there and $\tilde{f}$ has exactly one zero on each flow segment, so the zero set $Z={\tilde{f}}^{-1}(0)$ is a smooth compact hypersurface and the flow projection defines a homeomorphism $Z\to L$ [F3, F4, F5].

(${\pi }_1$ of the smooth model.) The set $Z$ is a closed smooth hypersurface [F5]; since the flow collar is a homeomorphism onto a collar of $L$, $Z$ is compact and connected for a sufficiently small collar, and the flow projection is a homeomorphism $Z\to L$ [F4]. By [F6] the closed smooth manifold $Z$ has the homotopy type of a finite CW complex, and a homotopy equivalence induces the isomorphism ${\pi }_1(Z,z)\cong {\pi }_1(X,\ast )$ for the finite CW complex $X$; the fundamental group of a finite CW complex is finitely generated [F7]. Hence ${\pi }_1(Z,z)$ is finitely generated, and the homeomorphism $Z\cong L$ transfers finite generation to ${\pi }_1(L,x)$, which is therefore finitely generated.

The compact leaf $L$ has a smooth compact hypersurface model $Z$ homeomorphic to it, the fundamental group of $Z$ is finitely generated, and homeomorphism invariance of ${\pi }_1$ gives the same for ${\pi }_1(L,x)$ [F7]. The Axiom of Choice is used through the finite-CW model [F6] and its countable-choice consumption, which full AC supplies by [F8].