Salem number stretch factors and totally real fields arising from Thurston's construction (Q2211886)

A homeomorphism \(\phi\) from a closed orientable surface \(S_{g}\), of genus \(g\), to itself is said to be pseudo-Anosov if there is a pair \((\mathcal{F}_{s},\mathcal{F}_{u})\) of transverse measured foliations of \(S_{g}\) in which \(\phi\) stretches \(\mathcal{F}_{s}\) by a real number \(\lambda >1\), and contracts \(\mathcal{F}_{u}\) by a factor of \(\lambda^{-1}\). Such a number \(\lambda\), called the stretch (or the dilatation) factor of \(\phi\), is an algebraic unit of degree at most \(6(g-1)\) [\textit{W. P. Thurston}, Bull. Am. Math. Soc., New Ser. 19, No. 2, 417--431 (1988; Zbl 0674.57008)]. Among the known constructions of pseudo-Anosov maps, Thurston provided a one which describes pseudo-Anosov automorphisms as products of Dehn twists around simple closed curves that divide the related surfaces into disks. Let \(\mathcal{F}\) designate the set of stretch factors of pseudo-Anosov maps arising from this construction. According to [\textit{P. Hubert} and \textit{E. Lanneau}, Duke. Math. J. 133, No. 2, 335--346 (2006; Zbl 1101.30044)], if \(\lambda \in \mathcal{F}\), then the algebraic integer \(\lambda +\lambda^{-1}\) is totally real, i. e., all (Galois) conjugates of \(\lambda +\lambda^{-1}\) are real. Consequently, every member of \(\mathcal{F}\) is a totally real unit of degree at least \(2\), or is a \(j\)-Salem number \((j\in \mathbb{N})\) whose imaginary conjugates belong to the unit circle. Recall that a \(j\)-Salem number \(\tau\) is an algebraic integer having \(j\) conjugates, including \(\tau\), with modulus greater than \(1\), and at least a conjugate with modulus \(1\). This is a generalization of the classical notion of Salem numbers, where \(j=1\) and \(\tau\) is a positive real number; the set of Salem numbers is, traditionally, denoted by \(\mathcal{T}\). In the paper under review, the author investigates the set \(\mathcal{F\cap T}\). Specifically, he shows that every Salem number has a power that belongs to \(\mathcal{F}\) (Theorem A), and any totally real number field of degree at least \(2\) has a primitive element of the form \(\lambda +\lambda^{-1}\), where \(\lambda \in \mathcal{F\cap T}\) \ (Theorem B). In fact, this second result asserts that \(\lambda \in \mathcal{F}\), and from its proof one can obtain that \(\lambda \in \mathcal{T}\), with minimal polynomial \(\prod\limits_{k=1}^{d}(x^{2}-(\alpha_{k}^{2}-2)x+1)\), where \(\alpha_{1},\dots,\alpha_{d}\) are the conjugates of the totally real Pisot unit \(\alpha\) defined in Lemma 7.5. Finally, notice that Theorem 7.1, used in the proof of Theorem B, is a corollary of a result of Pisot, saying that any real number field of degree at least \(2\) has a primitive element which is a Pisot unit (see for instance [\textit{M. J. Bertin} et al., Pisot and Salem numbers. Basel: Birkhäuser Verlag (1992; Zbl 0772.11041)], or [\textit{M. J. Bertin} and the reviewer, C. R., Math., Acad. Sci. Paris 353, No. 11, 965--967 (2015; Zbl 1332.11093)], where this theorem is extended to certain imaginary number fields).