Locating elementary argument for leaf volume bound of codimension one compact foliations

Question

Where in Reeb's thesis (or other expository reference) is the fact claimed by the following two blurbs proved? These both refer to Reeb's thesis.

Edwards-Millett-Sullivan's Foliations with all leaves compact says the following in the introduction:

THE PURPOSE of this paper is to present some information about the following Question:

If $M$ is a compact manifold foliated by compact submanifolds (everything smooth), is there an upper bound on the volume of the leaves?

...

Concerning the known cases of the Question, if the leaves have condimension 1 then the bound exists by a relatively elementary argument [12].

Also, Epstein's Foliations with all leaves compact says

Let M be a foliated manifold with each leaf compact. At first sight it seems reasonable to expect that for any compact subset K of M, the set of leaves meeting K should have bounded volumes (see § 3 for the the definition of the "volume" of a leaf). We shall talk of this belief as the conjecture. The investigation of this question was started by Reeb [10]. Reeb showed that in codimension one the conjecture was true.

Theorem 4.2 of the Epstein paper states that for a compact leaf in a foliated space, boundedness of the leaf volume on a neighborhood is equivalent to finiteness of the holonomy group, and also that these conditions imply the leaf has arbitrarily small saturated neighborhoods of compact leaves. So I think the two papers are referring to the parts just before and including Theorem (B, III, 3) of Reeb's thesis, but he also has this worrisome description of part B.

Dans le chapitre B nous étudions les trois sortes de théorèmes de stabilité qu'on peut formuler lorsqu'on suppose l'existence d'une feuille compacte $V_q$, à groupe de Poincaré fini.

Theorem (B, III, 3): (B, m, 3). THÉORÈME. — En particulier si $V_n$ est compact et si toutes les feuilles sont compactes, $V_n/p$ est aussi compact, et est homéomorphe an cercle $S$, ou à un intervalle fermé. Si toutes les feuilles sont bilatères, $V_n / p$ est une variété à une dimension (et seulement dans ce cas). Donc, si $V_n$ est compact, il y a o ou 2 feuilleseompacles uniiatères. ...

I can't seem to find another source explicitly justifying the claims at the beginning of this post. E.g., the global Reeb stability theorem assumes the existence of a leaf with finite fundamental group. Would somebody knowledgeable about these things mind either recommending an English exposition on the elementary argument, or explain the elementary argument, or otherwise point out explicitly where the elementary argument occurs in Reeb's thesis? I think I'm having a hard time finding this for myself because I'm just beginning to learn about foliation theory, and also I've never learned French.

Answer 1

My French hasn't improved, so I still haven't verified exactly what's in Reeb's thesis. I do think [Candel-Conlon] has the required arguments too, however:

Pablo Lessa in Reeb stability and the Gromov-Hausdorff limits of leaves in compact foliations writes the following in the introduction.

The Reeb local stability theorem [Ree47, Theorem 2] states that if the fundamental group of a compact leaf in a foliation is finite then all nearby leaves are finite covers of it. It’s apparent from the proof that, besides compactness, the key property of the leaf which yields stability isn’t finiteness of its fundamental group, but finiteness of its holonomy group. This gives rise to the standard generalization given for example in [CLN85, pg. 70].

Then I looked and I believe I found that in the proof of the global Reeb stability theorem from [Candel-Conlon] the only purpose of the the finite fundamental group assumption is to immediately conclude that the holonomy is finite.

[Candel-Conlon, Theorem 6.1.5]: Let $(M, \mathcal{F})$ be a compact, connected, transversely orientable, foliated manifold of codimension one. If there is a compact leaf $L$ having finite fundamental group (GS: there is apparently no harm at all in improving this assumption to the assumption that the compact leaf $L$ has finite holonomy group), then every leaf is homeomorphic to $L$ and $M$ is homeomorphic either to $L \times [0, 1]$, foliated as a product, or to the total space of a fiber bundle $p : M \to S^1$ having the leaves of $\mathcal{F}$ as fibers.

[Candel-Conlon Theorem 2.3.12] states that for arbitrary foliated manifolds, leaves having trivial holonomy are generic (which here means their union contains a countable intersection of open, dense subsets, so is in particular non-empty). So the assumption that a codimension one smooth foliation in a compact manifold has all compact leaves does lead to the conclusion of [Candel-Conlon, Theorem 6.1.5].

This together with Epstein's Theorem 4.2 also gives you boundedness of the leaf volume.

Lessa, Pablo, Reeb stability and the Gromov-Hausdorff limits of leaves in compact foliations, Asian J. Math. 19, No. 3, 433-464 (2015). ZBL1323.57017.

Candel, Alberto; Conlon, Lawrence, Foliations I, Graduate Studies in Mathematics. 23. Providence, RI: American Mathematical Society (AMS). xiv, 402 p. (2000). ZBL0936.57001.