Geometry and topology of surfaces (Q2663774)

The book has 12 chapters based on lectures of a graduate course given by the author at the ETH Zürich in 2018. The basic objects considered are hyperbolic structures on a closed orientable surface \(\Sigma\), the mapping class group \(\mathrm{MCG}(\Sigma)\) of isotopy classes of orientation-preserving diffeomorphisms of \(\Sigma\), the Teichmüller spaces \(\mathcal{T}(\Sigma)\) of hyperbolic structures on \(\Sigma\) up to isometries isotopic to the identity, and finally the space of projective measured foliations \(\mathcal{PMF}(\Sigma)\) of \(\Sigma\). As explained in the well-written short and efficient introduction, a general setting and framework for the book is as follows. Denoting by \(\mathcal{S}(\Sigma)\) the set of isotopy classes of non-contractible embedded circles (simple closed curves) in \(\Sigma\), the isotopy class of a diffeomorphism \(f\) of \(\Sigma\) is determined by its action on \(\mathcal{S}(\Sigma)\). The natural map \(\mathcal{T}(\Sigma)\to\mathbb R_{>0}^{\mathcal{S}(\Sigma)}\) (i.e., with the lengths of the geodesics of a specific hyperbolic surface \(\Sigma\) as coordinates) is a proper embedding, its projective version \(\mathcal{T}(\Sigma)\to\mathbb P(R_{>0}^{\mathcal{S}(\Sigma)})\) is still an embedding but no longer proper, and the compactification of the image of \(\mathcal{T}(\Sigma) \cong\mathbb R^{6g-6}\) in \(P(R_{>0}^{\mathcal{S}(\Sigma)})\) is a closed ball \(\mathcal{T}(\Sigma) \cup\mathcal{PMF}(\Sigma) \cong D^{6g-6}\). Finally, the mapping class group \(\mathrm{MCG}(\Sigma)\) acts on the closed ball \(\mathcal{T}(\Sigma) \cup\mathcal{PMF}(\Sigma)\), and by the classical Brouwer fixed point theorem each mapping class \(f\) has a fixed point: if this fixed point is in \(\mathcal{T}(\Sigma)\) then \(f\) is isotopic to a periodic diffeomorphism, otherwise \(f\) is either reducible (along a family of disjoint simple closed curves) or pseudo-Anosov, and this gives the Nielsen-Thurston classification of diffeomorphisms of \(\Sigma\). A pseudo-Anosov map \(f\) comes with a stretch factor \(\lambda (f) > 1\), and ``we will put a special emphasis on studying pseudo-Anosov maps with small stretch factors''. The 12 chapters of the book are the following: 1. Étude on the flat case: \(S^1 \times S^1\). 2. Plane hyperbolic geometry (a highlight here is a result of \textit{A. Basmajian} [Am. J. Math. 115, No. 5, 1139--1159 (1993; Zbl 0794.30032)] giving an explicit formula for the length of the boundary curve of a hyperbolic surface with one geodesic boundary component in terms of the orthospectrum of the boundary curve). 3. Simple closed geodesics and systoles. 4. Fenchel-Nielsen coordinates. 5. Dehn twists. 6. Normal generators for mapping class groups (proving in particular Dehn's theorem that the mapping class group is generated by Dehn twists along simple closed curves, and discussing a result of Lanier and Margalit that a pseudo-Anosov map with a small stretching factor normally generates the mapping class group). 7. Measured foliations. 8 The \((9g-9)\)-theorem for measured foliations. 9. Compactification of Teichmüller space. 10. Classification of mapping classes (the Nielsen-Thurston classification). 11. Perron-Frobenius theory (the stretch factor of a pseudo-Anosov diffeomorphism is a Perron-Frobenius eigenvalue, i.e. the largest eigenvalue of a certain matrix associated to the diffeomorphism). 12. Pseudo-Anosov maps with small stretch factor. Concluding, there is quite a lot of material on the just about 70 pages of the book (with a strong taste for hyperbolic trigonometry), a list of 62 references and a useful guide to the recent literature (the classical text in the area is the book by \textit{A. Fahti}, \textit{F. Laudenbach} and \textit{V. Poénaru} [Travaux de Thurston sur les surfaces. Seminaire Orsay. Paris: Société Mathématique de France (1979; Zbl 0406.00016)], for mapping class groups there is the recent book by \textit{B. Farb} and \textit{D. Margalit} [A primer on mapping class groups. Princeton, NJ: Princeton University Press (2011; Zbl 1245.57002)]).