Recent zbMATH articles in MSC 30Fhttps://zbmath.org/atom/cc/30F2023-05-08T18:47:08.967005ZWerkzeugOn chromatic numbers of space-times: open problemshttps://zbmath.org/1507.050352023-05-08T18:47:08.967005Z"Kosheleva, Olga"https://zbmath.org/authors/?q=ai:kosheleva.olga-m"Kreinovich, Vladik"https://zbmath.org/authors/?q=ai:kreinovich.vladik-ya(no abstract)A congruence family for 2-elongated plane partitions: an application of the localization methodhttps://zbmath.org/1507.110942023-05-08T18:47:08.967005Z"Smoot, Nicolas Allen"https://zbmath.org/authors/?q=ai:smoot.nicolas-allenFor a nonnegative integer \(k\), \textit{G. E. Andrews} and \textit{P. Paule} [J. Number Theory 234, 95--119 (2022; Zbl 1484.11201)] considered the \(k\)-elongated plane partition function \(d_k(n)\) where
\[
\sum_{n=0}^{\infty}d_k(n)q^n=\prod_{n=1}^{\infty} \frac {(1-q^{2n})^k}{(1-q^n)^{3k+1}}, \quad |q|<1.
\]
\(d_k(n)\) generalizes the unrestricted partition function since \(d_0(n)=p(n)\). Moreover, \(k\)-elongated plane partitions are also generalizations of partition diamonds developed by \textit{G. E. Andrews} et al. [Adv. Appl. Math. 27, 231--242 (2001; Zbl 0992.05017)].
Andrews and Paule [loc. cit.] conjectured that, for positive integers \(n\) and \(\alpha\), \(8n\equiv 1 \bmod 3^{\alpha}\) implies that \(d_2(n)\equiv 0 \bmod 3^{\alpha}\).
In the paper under review the author proves the refined form \(d_2(n)\equiv 0 \bmod 3^{2\lfloor \alpha/2\rfloor+1}\) of the conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of \(\mathbb{Z}[X]\).
Reviewer: Mihály Szalay (Budapest)Boundary of the Gothen componentshttps://zbmath.org/1507.140502023-05-08T18:47:08.967005Z"Ouyang, Charles"https://zbmath.org/authors/?q=ai:ouyang.charles"Tamburelli, Andrea"https://zbmath.org/authors/?q=ai:tamburelli.andreaThe aim of this paper is to describe a new phenomenon about the length spectrum compactification of the \(\mathrm{Sp}(4,\mathbb{R})\)-maximal representations of the fundamental group of a closed surface. In this paper, they show that all Gothen components \(\mathcal{G}(k)\) for all \(1\leq k\leq2g-3\), up to taking a quotient by a compact factor, share the same boundary in their length spectrum topology. In a previous paper [\textit{C. Ouyang} and \textit{A. Tamburelli}, ``Length spectrum compactification of the \(\mathrm{SO}_0(2,3)\)-Hitchin component'', Preprint, \url{arXiv:2010.03499}], the authors used a pseudo-Riemannian point of view to define the length spectrum of an \(\mathrm{Sp}(4,\mathbb{R})\)-Hitchin representation and describe the boundary of this component (modulo an \(S^1\)-action) as projectivized mixed structures, that is hybrid geometric objects on the surface that are measured laminations on some collection of incompressible subsurfaces and flat metrics of finite area induced by meromorphic quartic differentials on the complement.
The authors show that the Hitchin component and all Gothen components share the same boundary in their length spectrum compactification. The main idea behind the proof of this result lies in a comparison between the induced metric on the maximal surface associated to a maximal representation and the flat metric \(|q|^{\frac{1}{2}}\) with cone singularities, where \(q\) is the holomorphic quartic differential in the Hitchin base. They show that for any diverging sequence in \(\mathcal{G}(k)\), the length spectra of these two metrics after a suitable renormalization converge to that of mixed structures which enjoy the same decomposition into subsurfaces and coincide on their non-laminar part. As an application of their estimates and the work of \textit{S. Dai} and \textit{Q. Li} [Math. Ann. 376, No. 3--4, 1225--1260 (2020; Zbl 1448.14030)], they also describe the length spectrum for the induced metrics on the unique minimal surfaces in \(\mathrm{Sp}(4,\mathbb{R})/U(2)\).
The paper is organized as follows. Section 0 is an introduction to the subject. Section 1 deals with background material. Sections 2 is devoted to boundary of the Gothen components, and Sections 3 to induced metrics on the minimal surfaces.
Reviewer: Ahmed Lesfari (El Jadida)The heights theorem for infinite Riemann surfaceshttps://zbmath.org/1507.300102023-05-08T18:47:08.967005Z"Šarić, Dragomir"https://zbmath.org/authors/?q=ai:saric.dragomirAn integrable holomorphic quadratic differential on an arbitrary hyperbolic Riemann surface \(X\) defines a measured foliation by horizontal trajectories (that are geodesics for the corresponding flat metric on \(X\)). Lifting the leaves of this foliation to the universal cover of \(X\) and replacing them by geodesics with the same end points defines an associate measured lamination (geodesic for the hyperbolic metric) on \(X\). This construction gives a \textit{horizontal} map from the space of integrable quadratic differentials on \(X\) to the space of measured laminations on \(X\).
By a celebrated result of \textit{J. Hubbard} and \textit{H. Masur} [Acta Math. 142, 221--274 (1979; Zbl 0415.30038)], this map is an homeomorphism in the case of closed hyperbolic surfaces. The first result of this paper is that this map is injective for arbitrary hyperbolic Riemann surfaces, extending a result of Strebel for the disk.
Using the horizontal measured foliation, one can define the \textit{height} of a homotopy class of a simple closed curve by measuring the minimal vertical variation along curves in the class, or equivalently, by intersecting the geodecic representative with the measured lamination associated to the horizontal foliation.
A theorem of \textit{A. Marden} and \textit{K. Strebel} [Acta Math. 153, 153--211 (1984; Zbl 0577.30038)] states that the heigths determine entirely the quadratic differential, for parabolic Riemann surfaces (an arbitrary open Riemann surface is parabolic is there is no Green's function). The author proves that this result holds for Riemann surfaces with fundamental group of the first type (i.e., with limit set equal to the whole circle \(\partial\mathbb H\)), which excludes surfaces containing funnels or upper half-planes. In such surfaces a measured lamination is determined by its intersection with all simple closed geodesics.
Several other results are deduced for Riemann surfaces having a bounded pants decomposition or Riemann surfaces having a sequence of simple closed geodesics of length going to 0.
Reviewer: Elise Goujard (Talence)Counting geodesics, Teichmüller space, and random hyperbolic surfaceshttps://zbmath.org/1507.300112023-05-08T18:47:08.967005Z"Wolpert, Scott A."https://zbmath.org/authors/?q=ai:wolpert.scott-aThis interesting overview article is about the study of geodesics on hyperbolic surfaces of finite area. The author shows how the understanding of the behavior, invariants and length estimates of geodesics has lead to important insights into the geometry of hyperbolic surfaces and the moduli space of Riemann surfaces.
He starts by explaining how lengths of geodesics and corresponding twist parameters can be related to formulas for Teichmüller and moduli space and then gives a short overview over the Weil-Petersson geometry, the random geodesic metric and finally Mirzakhani's method of integration over moduli space. This method has lead to a number of important applications that have provided volume formulas for moduli spaces, asymptotic counts by length for simple closed geodesics and length estimates for geodesics on random hyperbolic surfaces.
Reviewer: Björn Mützel (St. Petersburg, FL)A topological characterization of the strong disk property on open Riemann surfaceshttps://zbmath.org/1507.300122023-05-08T18:47:08.967005Z"Abe, Makoto"https://zbmath.org/authors/?q=ai:abe.makoto"Nakamura, Gou"https://zbmath.org/authors/?q=ai:nakamura.gou"Shiga, Hiroshige"https://zbmath.org/authors/?q=ai:shiga.hiroshigeSummary: In this paper, we give a topological characterization of a subdomain \(G\) of an open Riemann surface \(R\) which has the strong disk property. Namely, we show that the domain \(G\) satisfies the strong disk property in \(R\) if and only if the canonical homomorphism \(\pi_1(G) \to \pi_1(R)\) is injective.A high-genus asymptotic expansion of Weil-Petersson volume polynomialshttps://zbmath.org/1507.320012023-05-08T18:47:08.967005Z"Anantharaman, Nalini"https://zbmath.org/authors/?q=ai:anantharaman.nalini"Monk, Laura"https://zbmath.org/authors/?q=ai:monk.lauraSummary: The object under consideration in this article is the total volume \(V_{g, n}(x_1, \dots, x_n)\) of the moduli space of hyperbolic surfaces of genus \(g\) with \(n\) boundary components of lengths \(x_1, \dots, x_n\), for the Weil-Petersson volume form. We prove the existence of an asymptotic expansion of the quantity \(V_{g, n}(x_1, \dots, x_n)\) in terms of negative powers of the genus \(g\), true for fixed \(n\) and any \(x_1, \dots, x_n \geq 0\). The first term of this expansion appears in the work of \textit{M. Mirzakhani} and \textit{B. Petri} [Comment. Math. Helv. 94, No. 4, 869--889 (2019; Zbl 1472.57028)], and we compute the second term explicitly. The main tool used in the proof is Mirzakhani's topological recursion formula, for which we provide a comprehensive introduction.
{\copyright 2022 American Institute of Physics}Lengths spectrum of hyperelliptic componentshttps://zbmath.org/1507.370442023-05-08T18:47:08.967005Z"Boissy, Corentin"https://zbmath.org/authors/?q=ai:boissy.corentin"Lanneau, Erwan"https://zbmath.org/authors/?q=ai:lanneau.erwanLet \(S_g\) be a genus-\(g\) closed topological surface. The expansion factor \(\lambda(\phi)>0\) of a pseudo-Anosov map \(\phi\) on \(S_g\) measures the exponential growth rate of the lengths of the curves under iteration of \(\phi\). The lengths spectrum of the moduli space \(\mathcal{M}_g\) for the Teichmüller metric is the set of the logarithms of all expansion factors (when fixing the genus); this is a discrete subset of \(\mathbb{R}\). It is considered to be challenging to describe the set of expansion factors of pseudo-Anosov mappings.
In this paper, the authors propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. Recall that half-translation surfaces, an important feature of pseudo-Anosov mappings, are determined by a pair \((X,q)\), where \(X\) is a Riemann surface and \(q\) is a meromorphic quadratic differential. The authors write \((X,\omega)\), when \(q=\omega^2\) is the global square of a holomorphic 1-form, and refer to \((X,\omega)\) as a translation surface.
There are many interesting consequences of this approach; most notable is the computation of the systole of the Teichmüller geodesic flow restricted to many different connected components of the strata of Abelian and quadratic differentials. Among them, the hyperelliptic components comprise entirely hyperelliptic curves. This answers a question of \textit{B. Farb} [Proc. Symp. Pure Math. 74, 11--55 (2006; Zbl 1191.57015)], see Theorem A of the introduction.
On the description of the systole of moduli spaces there were scarce results, and all of them concerned some cases in small genera. Also, bounds have been given by the authors as well as by \textit{C. T. McMullen} [J. Topol. 8, No. 1, 184--212 (2015; Zbl 1353.37033)]
and \textit{R. C. Penner} [Proc. Am. Math. Soc. 113, No. 2, 443--450 (1991; Zbl 0726.57013)]; see further references within the paper. With the method proposed here, we have the first results on the description of the systole of moduli spaces.
Finally, the methods developed in this paper provide a way to describe the bottom of the lengths spectrum of the hyperelliptic components for any genus, see Theorem B in the introduction. A picture of that for small genera is also provided here.
Notably, the proofs and computations within the paper are such that can be performed without the help of a computer.
Reviewer: Ioannis D. Platis (Heraklion)Existence of closed geodesics through a regular point on translation surfaceshttps://zbmath.org/1507.370462023-05-08T18:47:08.967005Z"Nguyen, Duc-Manh"https://zbmath.org/authors/?q=ai:nguyen.duc-manh"Pan, Huiping"https://zbmath.org/authors/?q=ai:pan.huiping"Su, Weixu"https://zbmath.org/authors/?q=ai:su.weixuSummary: We show that on any translation surface, if a regular point is contained in some simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in \(\mathbb{RP}^1\). Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist on some translation surfaces. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. The proofs of our results use Apisa's classifications of periodic points and of \(\text{GL}(2, \mathbb{R})\) orbit closures in hyperelliptic components, as well as a recent result of \textit{A. Eskin} et al. [Ann. Math. (2) 188, No. 1, 281--313 (2018; Zbl 1398.32015)].Collected Works of William P. Thurston with Commentary: IV. The geometry and topology of three-manifolds: with a preface by Steven P. Kerckhoff. Edited by Benson Farb, David Gabai and Steven P. Kerckhoffhttps://zbmath.org/1507.570052023-05-08T18:47:08.967005Z"Thurston, William P."https://zbmath.org/authors/?q=ai:thurston.william-pVolume IV of William P. Thurston's Collected Works consists of his famous notes entitled \textit{The geometry and topology of three-manifolds}. These are notes from the course he gave at Princeton University during the academic year 1977--78 and the fall semester of 1978--79. They are written by Thurston with the help of Bill Floyd and Steve Kerckhoff. (In the Editor's Preface of the volume, written by Kerckhoff, we read: ``Thurston started the first year by writing up notes himself, but after a while he enlisted Bill Floyd and me to help. [\dots] What was written was a reasonable facsimile of the way Thurston was presenting his amazing mathematics to us, in real time.'') The plan consists of 13 chapters, but Chapters 10 and 12 are missing, and Chapter 11 contains only a short section on extensions of vector fields. The lectures corresponding to Chapter 7 were given by John Milnor. The overall subject is geometric structures and the topology of three-manifolds.
The notes require almost no prerequisites but they are not easy to read: one needs time, visualisation and imagination. On the other hand, reading every section and understanding the ideas behind it is always rewarding. It is not an exaggeration to say that hundreds of young mathematicians wrote a PhD thesis by trying to understand some section of these notes.
Chapter 1 is entitled \textit{Geometry of three-manifolds}. This chapter is an introduction to the whole set of notes. Thurston introduces several topological constructions of three-manifolds, starting with the operation of Dehn surgery. Then he discusses branched coverings of the sphere with branching locus a knot or link. Constructions include Heegaard splittings of three-manifolds, and manifolds obtained by the identification of faces of polyhedra. Then appears the complement of the figure-eight knot, an important hyperbolic manifold which will be shown later to be the simplest hyperbolic link complement. Later in the notes, this example plays a major role, in particular in the theory of hyperbolic volume. Constructions with ideal tetrahedra are hinted to. While discussing these topics, Thurston brings out a few fundamental conjectures and open questions related to decision problems in the construction of three-manifolds.
Chapter 2 is an introduction to non-Euclidean geometry: spherical (or elliptic) and hyperbolic, from a point of view which at the epoch was very original. After a glimpse into the mystery of elliptic geometry, Thurston explains several essential ideas about models and how to work with them, in particular with the sphere of imaginary radius (the hyperboloid model of hyperbolic geometry). The importance of right angled hexagons is highlighted, and Thurston derives the basic trigonometric formulae for triangles.
In Chapter 3, Thurston explains his theory of geometric structures on manifolds. He introduces two different notions, \(\mathcal{G}\)-manifolds and \((G, X)\)-manifolds. The latter ones are equipped with a developing map and a holonomy representation, a reformulation in Thurston's way and a generalization of ideas of Ehresmann, which he had already encountered in his work on foliations. He states the conjecture that every closed three-manifold with finite fundamental group carries an elliptic geometry, pointing out that this conjecture is stronger than the Poincaré conjecture.
Chapter 4 is entitled \textit{Hyperbolic Dehn surgery}. A parametrisation of ideal tetrahedra is introduced, based on a parametrization of Euclidean triangles, which Thurston will use in his constructions of three-manifolds involving gluing ideal tetrahedra.
A hyperbolic structure on the figure-eight knot complement is described in this setting. The completion of hyperbolic structures obtained by gluing hyperbolic ideal tetrahedra is studied via the developing map. The generalized Dehn surgery invariant is introduced. Dehn surgery on the figure-eight knot is given special attention. Foliations, and in particular the notion of hyperbolic foliation, are examples of \(\mathcal{G}\)-structures. They appear at this point in the study of limits of simplices. Thurston is then led to the degeneration theory of hyperbolic structures in terms of \(\mathcal{G}\)-structures. The chapter ends with a classification of incompressible surfaces in the figure-eight knot complement.
Chapter 5 is entitled \textit{Flexibility and rigidity of geometric structures}. Thurston explains how the holonomy map parametrizes small variations of a geometric structure. Hyperbolic structures on pairs of pants and then on surfaces are classified, and Teichmüller space is introduced. Then Thurston passes to dimension three, where he starts by studying isometries of hyperbolic 3-space and representations in \(\mathrm{PSL}(2,\mathbb{C}\)). This leads him to a count of the dimension of the space of small deformations of a hyperbolic three-manifold. He then appeals to Mostow's rigidity theorem to show that any complete finite-volume hyperbolic manifold is determined by its volume. Using this result, together with techniques of hyperbolic Dehn surgery and of completion of incomplete structures, he obtains an infinite family of complete closed hyperbolic manifolds which are all different. While doing this, he gives a new geometric proof of Mostow's theorem. He then discusses the thin-thick decomposition of complete oriented hyperbolic three-manifolds of finite volume, based on the Margulis lemma, and he gives several versions of a theorem which he attributes to Jørgensen saying that the volume function, from the set of complete hyperbolic three-manifolds of finite volume onto the positive reals, is proper. Furthermore, for any positive real number \(C\), there exists a finite collection of complete hyperbolic three-manifolds of volume \(\leq C\) such that any other complete hyperbolic three-manifold of volume \(\leq C\) is obtained from one of these manifolds by hyperbolic Dehn surgery.
Chapter 6 is entitled \textit{Gromov's invariant and the volume of a hyperbolic manifold}. The volume of a hyperbolic manifold, which, by Mostow rigidity, is a highly powerful invariant, is again at the center of the discussion. The related Gromov norm is introduced. This is a pseudo-norm on the singular homology of a topological space which measures how efficiently multiples of a homology class may be represented by simplices. A theorem of Gromov says that for finite-volume hyperbolic manifolds, the norm of the fundamental class coincides with the hyperbolic volume times a constant which depends only on dimension. Thus, the Gromov norm becomes an invariant for this class of manifolds. In the same chapter Thurston introduces several variants of Gromov's definition, and he gives a new proof of the latter's result. By combining a strict and a relative version of Gromov's theorem, he proves that if a complete hyperbolic manifold of finite volume \(M_2\) is obtained topologically from another complete hyperbolic manifold of finite volume \(M_1\) by replacing certain cusps by solid tori, then the volume of \(M_1\) is strictly greater than the volume of \(M_2\). In other words, the volume strictly decreases by Dehn filling. Furthermore, Thurston presents a new proof of Mostow's rigidity theorem for hyperbolic three-manifolds, based on Gromov's result. He notes that the proof will work for \(n\)-dimensional manifolds if one knows that the unique ideal \(n\)-simplex of maximal value is the regular one. Then follow discussions of manifolds with boundaries, of ordinals, and of commensurability of discrete subgroups of isometries of \(\mathbb{H}^n\). The chapter contains several examples where one can compute volumes of hyperbolic link complements, examples of manifolds with symmetries, and a discussion of limits of volumes. Other results of special interest regarding volume that Thurston presents include the fact that the set of volumes of hyperbolic three-manifolds is a closed non-discrete subset of the real line, and that it is well-ordered. Furthermore, the volume function of hyperbolic three-manifolds is finite-to-one. From this, he deduces that the set of volumes of hyperbolic three-manifold is indexed by countable ordinals and has the ordinal type of \(\omega^{\omega}\). Furthermore, for any non-compact hyperbolic three-manifold of finite volume, there exists an infinite sequence of non-compact hyperbolic three-manifolds whose volume is strictly less than that of the original manifold, that approximate it in the sense of pointwise convergence of the representations of the fundamental groups in \(\mathrm{PSL}(2,\mathbb{C})\), and whose volume converges to the volume of the original manifold. Furthermore, Thurston showed that a convergent sequence of real numbers consisting of values of volumes of hyperbolic manifolds that are link complements has always an increasing subsequence.
Chapter 7, based on lectures by John Milnor, is entitled \textit{Computation of volume}, and it is concerned with volumes of hyperbolic ideal polyhedra. Milnor introduces the Lobachevsky function, discussing its analytic properties, and he formulates two conjectures on number-theoretic properties of this function. He then shows how this function arises in computations of volumes of simplices and ideal simplices. This is based on work of Lobachevsky, where the function appears in a slightly different form. He then discusses examples that arise from knots and links complements (essentially the figure-eight knot and the Whitehead link).
The next two chapters are concerned with Kleinian groups, that is, discrete subgroups of isometries of hyperbolic 3-space.
Chapter 8 is entitled \textit{Kleinian groups}. The basic notions of limit set and domain of discontinuity are introduced. Thurston discusses then properties of convexity and compact-convexity for hyperbolic manifolds, and their characterization in terms of the developing map. Geometrically finite groups, whose action has a finite-sided fundamental polyhedron, are studied. Section 4 of this chapter starts with a proof of a theorem of Ahlfors for such groups, which says that the limit set is either of full or zero measure. In the case of full measure, the action of the group on the boundary at infinity is ergodic. In the same section, Thurston introduces the convex core of an action, as the convex hull of the limit set quotiented by the group action. This is a three-dimensional submanifold whose inclusion map in the ambient manifold is a homotopy equivalence. He then discusses the nearest point retraction of hyperbolic space, including the sphere at infinity, onto the convex hull of the limit set of a Kleinian group.
Thurston then discusses Fuchsian groups (Kleinian groups whose limit set is a geometric circle) and quasi-Fuchsian groups (Kleinian groups whose limit set is a topological circle) and their deformations. Quasi-Fuchsian groups give rise to manifolds homeomorphic to surfaces times an interval. Uncrumpled surfaces appear in the discussion. These are surfaces bent along geodesic laminations, embedded (a priori not necessarily injectively) in the manifold. A quasi-Fuchsian manifold is filled in with uncrumpled surfaces. It seems that the terminology ``uncrumpled surface'' completely disappeared later, to the advantage of ``pleated surface''. The boundary of the convex core of the action of a Kleinian group is a pleated surface. The geodesic laminations that appear as bending loci of pleated or uncrumpled surfaces are equipped with transverse measures, defined from the amount of bending. Thurston is led here to the development of the theory of geodesic and measured geodesic laminations on hyperbolic surfaces. In the meantime, train tracks are introduced, defining parameter spaces for measured geodesic laminations. Hyperbolic surfaces and geodesic laminations are then used in the analysis of cusps of hyperbolic three-manifolds. Thurston introduces the notion of geometrically tame end, which involves the fact that there exists a sequence of homotopic pleated surfaces that tend to infinity in that end. The last section of the chapter is entitled \textit{Harmonic functions and ergodicity.} Thurston proves there a theorem on the behavior of positive harmonic functions defined on the convex hull of geometrically tame ends, implying that the limit set of the action of the fundamental group of a tame end has measure 0 or 1, and that in the latter case the group acts ergodically on the two-sphere.
Chapter 9 is entitled \textit{Algebraic convergence}. It is concerned with the topology of the space of discrete groups of isometries of hyperbolic space. The chapter starts with the notion of limit of a sequence of discrete groups, and Thurston explains that there is more than one useful sense in which such a sequence can have a limit. Then, the notion of geometric convergence of a sequence of discrete groups reflects that of geometric convergence of a sequence of complete finite-volume hyperbolic manifolds. Algebraic and strong algebraic convergence are then introduced, with many examples and properties of the topologies they induce on spaces of groups, and of the hyperbolic manifolds that arise as limits. A comparison between different notions of convergence is made, and examples due to Jørgensen are discussed. The properties of geometric tameness and almost geometric tameness of representations are introduced. Thurston then starts a thorough study of the geometry of ends. Topologically, geometrically and almost geometrically tame ends are introduced, and in particular the question of the realizability of a lamination in a geometrically tame end. This also leads Thurston to a more detailed study of geodesic laminations. The piecewise linear structure of spaces of laminations is introduced, together with the action of mapping class groups on these spaces. The notion of rational depth of a measured lamination is introduced. Algebraic limits and strong algebraic limits are revisited, then the notion of realization of geodesic laminations for surface groups with extra cusps, with a digression on stereographic coordinates. Thurston proves a theorem of Sullivan giving several equivalent conditions for the ergodicity of the geodesic flow on a complete \(n\)-dimensional hyperbolic manifold, including the notion the divergence of a Poincaré series at the critical exponent.
Chapter 11 is called \textit{Deforming Kleinian manifolds by homeomorphisms of the sphere}. It consists of a single section, on extension of vector fields. It contains tensor calculus and many formulae. (It is not written in the usual Thurston style.) The volume's editor writes about this section: ``This was used in the latter part of the semester to extend isotopies from the boundary to the interior of geometric structures during the process of analyzing the space of such structures on certain 3-manifolds. There were other topics covered that semester but they did not make it into the notes.''
From a note written by Thurston and inserted in the volume under review between Chapters 10 and 11, we learn that the initial outline of the lectures corresponding to Chapter 11 was: The Riemann mapping theorem, parameterizing quasi-conformal deformations, extending quasi-conformal deformations of \(S^2_{\infty}\) to quasi-isometric deformations of \(H^3\), examples, conditions for the existance of limiting Kleinian groups. Likewise, we learn that the outline of the lectures corresponding to Chapter 12 was: Boundaries of Teichmüller space, classification of diffeomorphisms of surfaces, algorithms involving the mapping class group of surfaces.
Let me mention here that about the same time these notes were released, the volume \textit{Travaux de Thurston sur les surfaces}, covering part of the topics announced for Chapter 12, appeared. (The volume was published in 1979, and the corresponding seminar was held in 1976--77.) The other topics announced in Chapters 10 and 11 appeared later in various forms, in writings of Thurston, some of them with collaborators.
Chaper 13 is entitled \textit{Orbifolds}. Orbifolds are quotients of group actions that are properly discontinuous but not necessarily free (they may have fixed points). Thurston writes in the introduction: ``In the first place, such quotient spaces will yield a technical device useful for showing the existence of hyperbolic structures on many three-manifolds. In the second place, they are often simpler than three-manifolds tend to be, and hence they often give easy, graphic examples of phenomena involving three-manifolds. Finally, they are beautiful and interesting in their own right.'' The beauty of the examples that Thurston gives in this setting (like the beauty of the theories and examples that appear elsewhere in these notes) is an instance of his exceptional artistic and geometric vision.
The chapter starts with examples of quotient spaces of group actions, after which Thurston gives the definition of good and bad orbifold. He then develops the theory of covering orbifolds, of orbifold fundamental group, and of orbifolds with boundary. Then he makes a complete theory of two-dimensional orbifolds, in the Euclidean and non-Euclidean settings. Then comes the theory of tangent spaces to orbifolds, and fibrations between orbifolds, where knot and link complements and the other three-dimensional spaces that we saw in the preceding chapters appear again at the center of the discussion. The theory of reflection groups leads Thurston to an amazing proof of a theorem of Andreev on hyperbolic reflection groups, based on a constructive algorithm for the existence of circle packings on the sphere that are dual to some graphs satisfying certain simple combinatorial conditions. This is the origin of a whole theory of circle packings that Thurston outlined and which led to remarkable developments. Thurston introduces then the notions of Kleinian structures on orbifolds, and of Kleinian structure with nodes. Bending laminations and volumes are revisited. The chapter ends with a geometric compactification of the Teichmüller spaces of polygonal orbifolds and a geometric compactification for the deformation spaces of certain Kleinian groups.
The notes end with the sentence: ``To be continued \dots''
These notes constitute the most influential text ever written on the topology and geometry of three-manifolds. Soon after they were released, they dramatically transformed the field of geometric topology in the sense that they gave it a completely new and unpredicted direction which turned out to be highly fecund, providing the necessary tools and techniques for the field development in this direction. Today, these notes remain fresh and the ideas they contain constitute an inexhaustible source of inspiration.
I have only one complaint concerning the edition: the index does not correspond to the actual page numbers, and this is unfortunate. It is just a latex compilation error, but it makes the index not useful. The mistake is not due to the volume editor but to the final typesetter of the book publisher.
The Collected Works volume under review is edited by Steve Kerckhoff who wrote a very informative preface. I cannot but recommend to any geometer/topologist -- student or senior -- to acquire this volume, together with the three preceding volumes.
Reviewer: Athanase Papadopoulos (Strasbourg)The boundary at infinity of the curve complex and the relative Teichmüller spacehttps://zbmath.org/1507.570212023-05-08T18:47:08.967005Z"Klarreich, Erica"https://zbmath.org/authors/?q=ai:klarreich.ericaLet \(S\) be a surface of a finite genus with finitely many punctures. Let \(\mathcal{T}(S)\) denote the Teichmüller space of \(S\), which is the space of all equivalence classes of conformal structures of finite type on the surface \(S\). A conformal structure is of finite type if each puncture has a neighborhood conformally equivalent to a punctured disk.
Let \(\alpha\) be a homotopy class of simple closed curves on \(S\). Thin\(_{\alpha}\) denotes the region of \(\mathcal{T} (S)\); that is, Thin\(_{\alpha} = \{ \sigma \in \mathcal{T} (S) \mid \mathrm{ext}_{\sigma}(\alpha) \leq \epsilon \,\}\) for some fixed small \(\epsilon > 0\), where ext\(_{\sigma}(\alpha)\) is the extremal length of \(\alpha\). We note that the extremal length ext\(_{\sigma}(\alpha) = \sup_{\rho} \frac{(l_{\rho}(\alpha))^2}{A_{\rho}}\), where \(\sigma \in \mathcal{T}(S)\), \(\rho\) ranges over all metrics in the conformal class of \(\sigma\), \(l_{\rho}(\alpha)\) is the infimum of the length of \(\alpha\) with respect to \(\rho\) and \(A_{\rho}\) is the area of \(S\) with respect to \(\rho\).
In this paper, the author aims to describe the boundary at infinity of the electric Teichmüller space \(\mathcal{T}_{el} (S)\), which is obtained from \(\mathcal{T} (S)\) by collapsing every region Thin\(_{\alpha}\) to diameter 1. To put it more explicitly, the author shows the following theorem:
The boundary at infinity of \(\mathcal{T}_{el} (S)\) is homeomorphic to the space of minimal topological foliations on \(S\).
Furthermore, the author shows that this result holds for the curve complex \(\mathcal{C} (S)\); that is, the boundary at infinity of \(\mathcal{C} (S)\) is the space of minimal topological foliations on \(S\). We point out that the curve complex \(\mathcal{C} (S)\) is a simplicial complex whose vertices are homotopy classes of non-peripheral simple closed curves on the surface \(S\). Here, two vertices are connected by an edge if curves corresponding to these vertices are realized disjointly on the surface \(S\).
Reviewer: Ferihe Atalan (Ankara)