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In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof, with several variations at each stage, that uses only the statement of the geometrization theorem, basic hyperbolic geometry, and old results from combinatorial topology and computer science. For this result, we do not rely on normal surface theory, methods from geometric group theory, nor methods used to prove geometrization. The new result is that the homeomorphism problem is elementary recursive, i.e., that the computational complexity is bounded by a bounded tower of exponentials. This result relies on normal surface theory, Mostow rigidity, and bounds on the computational complexity of solving algebraic equations., 24 pages. This version is a substantial revision and expansion in response to referee comments

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston's geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al., Comment: 18 pages, 5 Figure, Special Issue "Groups, Geometry and Topology for Quantum Computations" in Quantum Reports

An erratum to this article is posted at https://gcc.episciences.org/page/errata This paper generalizes results of M. Moon on the fibering of certain compact 3-manifolds over the circle. It also generalizes a theorem of H. B. Griffiths on the fibering of certain 2-manifolds over the circle., Comment: Published in the journal of Groups, Complexity, Cryptology. Erratum to this article posted at https://gcc.episciences.org/page/errata

By Thurston's hyperbolization theorem, irreducible handlebody-knots are classified into three classes: hyperbolic, toroidal, and atoroidal cylindrical. It is known that a non-trivial handlebody-knot of genus two has a finite symmetry group if and only if it is atoroidal. The paper investigates the topology of cylindrical handlebody-knots of genus two that admit an unknotting annulus; we show that the symmetry group is trivial if the unknotting annulus is unique and of type $2$., Comment: 12 pages, 5 figures

From Thurston’s geometrization theorem to the Banach-Tarski paradox, spatial complexity can be enigmatic, due to singularities, immersions, infinities (e.g. Alexander’s wild knots, Antoine’s necklace, Ford circles, Hilbert curves). Measuring spatial complexity in already “complex” spatial settings constitutes a major challenge for future research, as it should combine algorithmics, computation and topology within a single whole. Further, completely homogeneous maps can still be complex if their complexity is measured by CP1, because topological differentiation (i.e. division in square cells) creates spatial complexity: a completely undifferentiated square map without cells defined on it has a different spatial complexity than a square map with cells identified in it. And yet, the same allocations of numbers to cells of binary maps can produce different configurations with different spatial complexities.

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston&rsquo, s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement C ( K ) = S 3 ( K &times, D 2 ) of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of S 3 branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk D 3 branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld&ndash, Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson&ndash, Thompson model).

Generalized Fibonacci numbers { F (g) i } are defined by the recurrence relation F (g) i+2 = gF(g) i+1 + F (g) i with the initial conditions F (g) 0 = 1, F (g) 1 = g. These numbers generater representations of natural numbers as a greedy expansions n = ∑k i=0 ei(n)F (g) i , with natural conditions on ei(n). In particular, when g = 1 we obtain the well-known Fibonacci numeration system. The expansions obtained by g > 1 are called representations of natural numbers in generalized Fibonacci numeration systems. This paper is devoted to studying the sets F (g) (e0, . . . , el), consisting of natural numbers with a fixed end of their representation in the generalized Fibonacci numeration system. The main result is a following geometrization theorem that describe the sets F (g) (e0, . . . , el) in terms of the fractional parts of the form {nτg}, τg = √ g 2+4−g 2 . More precisely, for any admissible ending (e0, . . . , el) there exist effectively computable a, b ∈ Z such that n ∈ F (g) (e0, . . . , el) if and only if the fractional part {(n + 1)τg} belongs to the segment [{−aτg}; {−bτg}]. Earlier, a similar theorem was proved by authors in the case of classical Fibonacci numeration system. As an application some analogues of classic number-theoretic problems for the sets F (g) (e0, . . . , el) are considered. In particular asymptotic formulaes for the quantity of numbers from considered sets belonging to a given arithmetic progression, for the number of primes from considered sets, for the number of representations of a natural number as a sum of a predetermined number of summands from considered sets, and for the number of solutions of Lagrange, Goldbach and Hua Loken problem in the numbers of from considered sets are established.

The vision and results of William Thurston (1946–2012) have shaped the theory of 3-dimensional manifolds for the last four decades. The high point was Perelman’s proof of Thurston’s Geometrization Conjecture which reduced 3-manifold topology for the most part to the study of hyperbolic 3-manifolds. In 1982 Thurston gave a list of 24 questions and challenges on hyperbolic 3-manifolds. The most daring one came to be known as the Virtual Fibering Conjecture. We will give some background for the conjecture and we will explain its precise content. We will then report on the recent proof of the conjecture by Ian Agol and Dani Wise.

Let Σ−3 be the connected sum of three real projective planes. We realize the Thurston compactification of the Teichmuller space Teich(Σ−3 ) as a simplex in P(R). First, we define a map L2 from Teich(Σ − 3 ) into P(R ) in terms of some geodesic-length functions. We then introduce the similar triangle flow on Teich(Σ−3 ) to control the ratios between these lengths, and show that L2 is an embedding. Finally, we study the natural extension of L2 to the Thurston boundary using a triangulation of the projective space of measured foliations.

For any Thurston map with exactly four postcritical points, we present an algorithm to compute the Weil-Petersson boundary values of the corresponding Thurston pullback map. This procedure is carried out for the Thurston map f ( z ) = 3 z 2 2 z 3 + 1 f(z)=\frac {3z^2}{2z^3+1} originally studied by Buff, et al. The dynamics of this boundary map are investigated and used to solve the analogue of Hubbard’s Twisted Rabbit problem for f f .

A brief survey of some recent results on projective and quasiprojective groups of low cohomological dimensions was presented by the second author at the conference in Hyderabad University in March 2015. This is a slightly expanded version of the talk. Part of the survey involves joint work with H. Seshadri and A. J. Parameswaran.

In this paper, we prove that the fundamental group of an orientable compact 3-manifold has finite decomposition complexity if Thurston's Hyperbolization Conjecture is true.

These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds"., 216 pages, minor corrections made

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

We present a refinement of the Gauss–Lucas theorem, following an idea of W.P. Thurston.

We prove the hyperbolization theorem for punctured torus bundles and two-bridge link complements by decomposing them into ideal tetrahedra which are then given hyperbolic structures, following Rivin's volume maximization principle., This is the version published by Geometry & Topology on 16 September 2006. Appendix by David Futer

We show that if M is a surface bundle over S^1 with fiber of genus 2, then for any integer n, M has a finite cover tilde(M) with b_1(tilde(M)) > n. A corollary is that M can be geometrized using only the `non-fiber' case of Thurston's Geometrization Theorem for Haken manifolds., Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper19.abs.html

Let H denote hyperbolic n-space, that is the unique connected simply connected Riemannian manifold of constant curvature −1. By a hyperbolic n-orbifold we shall mean a quotient H/Γ where Γ is a discrete group of isometries of H. If a hyperbolic n-manifold M is the totally geodesic boundary of a hyperbolic (n+1)-manifold W , we will say that M bounds geometrically. It was shown in [11] that if a closed orientable hyperbolic M4k−1 bounds geometrically, then η(M4k−1) ∈ Z. Closed hyperbolic 3-manifolds with integral eta are fairly rare – for example, of the 11, 000 or so manifolds in the census of small volume closed hyperbolic 3-manifolds, computations involving Snap (see [3]) rule out all but 41. (We refer the reader to [24] which contains the list of manifolds in the census with Chern-Simons invariant zero, as well as which of these have integral eta.) Hyperbolic 3-manifolds with totally geodesic boundary are fairly easily constructed given the Hyperbolization Theorem of Thurston [20], but to the authors’ knowledge, there was only one known prior example of a closed hyperbolic n-manifold (with n ≥ 3) which bounded geometrically, a somewhat ad hoc construction which appears in [18], based on a hyperbolic 4-manifold example due to Davis [4]. The difficulty is that almost nothing is known about hyperbolic manifolds in dimensions ≥ 4; some constructions exist (see [5], [6], [7]) but they do not appear to be sufficient to address this problem. This paper ameliorates this situation somewhat by providing a construction of examples in all dimensions. We show

Lorenz knots and links have been an area of research for over thirty years. Their study combines several fields of mathematics, including topology, geometry, and dynamical systems. The introduction of the Lorenz template permitted a dimension reduction, the definition of the one-dimensional Lorenz map, and the use of symbolic dynamics to code its orbits. Meanwhile, Thurston’s geometrization theorem implies that all Lorenz knots can be classified as torus knots, satellites, or hyperbolic knots. Based on a list of hyperbolic knots presented by Birman and Kofman, we define three families of Lorenz knots, compute their hyperbolic volume using the topology and geometry software SnapPy and use the results to present conjectures concerning their hyperbolicity.

The key result in the present paper is a direct analogue of the celebrated Thurston’s Theorem Douady and Hubbard (Acta Math 171:263–297, 1993) for marked Thurston maps with parabolic orbifolds. Combining this result with previously developed techniques, we prove that every Thurston map can be constructively geometrized in a canonical fashion. As a consequence, we give a partial resolution of the general problem of decidability of Thurston equivalence of two postcritically finite branched covers of $$S^2$$ (cf. Bonnot et al. Moscow Math J 12:747–763, 2012).

A compact, orientable 3-manifoldMis calledhyperbolicif intMadmits a complete hyperbolic structure (Riemannian metric of constant curvature − 1) of finite volume. Any hyperbolic 3-manifoldMis irreducible, and each component of∂Mis an incompressible torus. Letf:M→Nbe a proper, continuous map between hyperbolic 3-manifolds. By Mostow's Rigidity Theorem [8], iffis π1-isomorphic thenfis properly homotopic to a diffeomorphismg:M→Nsuch thatg| intM: intM→ intNis isometric. In particular, the topological type of intMdetermines uniquely the hyperbolic structure on intMup to isometry, so the volume vol (intM) of intMis well-defined. This Rigidity Theorem is generalized by Thurston[11, theorem 6·4] so that any proper, continuous mapf:M→Nbetween hyperbolic 3-manifolds with vol(intM) = deg(f) vol(intN) is properly homotopic to a deg(f)-fold coveringg:M→Nsuch thatg| intM: intM→ intNis locally isometric.

International audience; In the theory of rational maps an important role is played by matings. These are probably the best understood of all rational functions, but they are bizarre, and involve gluing dendrites together to get spheres carrying Peano curves. In the theory of Kleinian groups, there is a parallel construction, the construction of double limits, that is central to Thurston’s hyperbolization theorem for 3-manifolds that fiber over the circle with pseudo-Anosov monodromy. It also involves gluing dendrites and Peano curves. Clearly these two constructions form one entry of the Sullivan dictionary. This article attempts to spell out the similarities and differences.

Rees-Shishikura's theorem plays an important role in the study of mating in polynomials. It promotes Thurston's combinatorial equivalence to a semi-conjugacy. In this work we restate and reprove Rees-Shishikura's theorem in a more general form, which can then be applied to a wider class of postcritically finite branch coverings. We provide an application of the restated theorem., Le théorème de Rees-Shishikura joue un rôle important dans l’étude des accouplements de polynômes. Il permet d’obtenir une semi-conjugaison à partir d’une équivalence combinatoire de Thurston. Dans ce travail, nous reformulons et redémontrons ce théorème dans un cadre plus général. Cette nouvelle version du théorème est applicable à une classe plus large de revêtements ramifiés posteritiquement finis. Nous en fournissons un exemple à la fin de notre article.

Last year, the Zentralblatt fur Mathematik and the Jahresbericht der DMV began a collaboration to review again classical books and papers and the effect that they have had or could have had on the development of specific mathematical branches from today’s point of view. It has turned out that these renewed reviews build a category of articles of its own and that they cannot in general be considered as book reviews or historical articles. For this reason we introduce the new category “Classics Revisited” and shall try to have one article of this kind in each issue. We begin with a new review by Jean-Pierre Otal of William Thurston’s 1982 paper on “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry”. This research announcement did not only contain his Geometrization Conjecture and his Hyperbolization Theorem for so called Haken manifolds but also a list of 24 of at that time open problems, of which 22 had been solved by 2012. Only few papers have been of such a strong influence and inspiration. Bosons are particles in quantum mechanics which do not obey the Pauli principle. If a large fraction of particles in a Bose fluid occupies the same quantum state, the so called Bose-Einstein condensation occurs and leads to physically remarkable phenomena like superfluidity. This means that the viscosity of a fluid vanishes completely at low temperatures. Robert Seiringer’s survey article on “The excitation spectrum for Bose fluids with weak interaction” reviews recent progress in understanding underlying mathematical models. Mass aggregation phenomena like e.g. polymerisation may be modelled by means of Smoluchowski’s coagulation equation. Barbara Niethammer’s survey article addresses the key question in understanding the large time behaviour of solutions

We prove hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance form the boundary of the convex core for points in the thick part. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves. We use this inflexibility to give a new proof of the convergence of pseudo-Anosov double-iteration on the quasi-Fuchsian space of a closed surface, and the resulting hyperbolization theorem for closed 3-manifolds that fiber over the circle with pseudo-Anosov monodromy., 52 pages. Final version. Appeared in Journal of Topology. Material in original version on inflexibility of hyperbolic cone-manifolds has been rewritten into a new paper with identifier arXiv:1412.4635

The notion of an orbifold is a generalization of the notion of a manifold which appears naturally in the context of properly discontinuous nonfree actions of groups on manifolds. Orbifolds were first invented by Satake [Sat57] in the 1950s under the name of V-manifolds. They were reinvented under the name of orbifolds by Thurston in the 1970s as a technical tool for proving the Hyperbolization Theorem. Our discussion of orbifolds follows mainly [Thu81, Chapter 13] and [Sco83a]. For a more detailed discussion of foundational issues of orbifold theory, we refer the reader to [Rat94, Chapter 13]. Note that many definitions that we take for granted in manifold theory are rather mysterious in the category of orbifolds. For instance, it is unclear what the correct definition of characteristic classes of orbifolds is (compare [Kaw78] and [HH90]). Even if one can define the Euler characteristic of an orbifold, it is unclear what the Betti numbers are. We will define the fundamental group of an orbifold, but it is far from obvious what the correct definition of higher homotopy groups, etc., is. All in all, it seems that orbifolds live in the grey area between the “good old manifold kingdom” and the dark world populated by the “wild creatures” of Alain Connes’ noncommutative geometry.

In this section we discuss various conjectures related to Thurston’s Hyperbolization Theorem. Our discussion is far from being comprehensive. We refer the interested reader to Kirby’s list of problems [Kir97] for more details.

Finally, we can present an outline of the proof of Thurston's Hyperbolization Theorem. Our discussion mainly follows [Mor84]. By Theorem 8.36, it is enough to prove the Hyperbolization Theorem for a finite covering of M. Thus we will consider only compact orientable pared manifolds (M, P), which contain orientable superincompressible surfaces and where \(P=\partial M\) is the designated parabolic locus. Thurston's theorem is proven by induction on levels of the Haken hierarchy of M. It turns out that the main problem in proving Theorem 1.42 is the last step of the induction.

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We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To illustrate these techniques, we also compute the Thurston polytopes of several specific link complements., Revised Section 3: improved exposition and corrected statement of Lemma 3.6. Also other minor revisions throughout

Given a continuous $l$-modal map $g$ of the interval $[0,1]$ we prove the existence of a polynomial $P$ with modality $\leq l$ such that $g$ is strongly semi-conjugate to $P$ in $[0,1]$. This is an improvement of a result in [4]. We do a modification on the Thurston operator in order to control the semi-finite combinatorial case. It turns out that all the essential attractors of $P$ have the same local topological type as those of $g$. This allows to construct the strong semi-conjugacy. We also present some examples agreeing with the results.

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1--> pi_1(S) --> L_G --> G -->1 we prove that if L_G is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a "Schottky subgroup" of MCG, the converse is true as well; a semidirect product of pi_1(S) by a free group G is therefore word hyperbolic if and only if G is a Schottky subgroup of MCG. The special case when G=Z follows from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance: sufficiently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup., Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper5.abs.html

Let $\Sigma$ be a hyperbolic link with $m$ components in a 3-dimensional manifold $X$. In this paper, we will show that the moduli space of marked hyperbolic cone structures on the pair $(X, \Sigma)$ with all cone angle less than $2\pi /3$ is an $m$-dimensional open cube, parameterized naturally by the $m$ cone angles. As a corollary, we will give a proof of a special case of Thurston's geometrization theorem for orbifolds., Comment: 29 pages

In this note we sketch our computation of the group A 2 of cobordism classes of orientation-preserving diffeomorphisms of closed, oriented surfaces. See Sect. 2 for precise definitions. This computation was made independently and a little earlier by Bonahon [2]. Our development follows in part the same lines, extending approaches to related questions by Scharleman [13] and Johannson and Johnson [10], who were the first to apply the characteristic manifold theory of Johannson [8] and Jaco and Shalen [7] to questions about surface diffeomorphisms. Bonahon's approach depends crucially on the Mostow Rigidity Theorem and the unpublished and as yet inaccessible Hyperbolization Theorem of Thurston. In contrast, we get by with the G-Signature Theorem (in dimension two, a classical formula) and some number theory. For this reason we have belatedly decided to publish our own approach. Our main contribution is a proof, independent of deep three-dimensional geometry, that an orientation-preserving periodic surface diffeomorphism which bounds as a diffeomorphism also bounds periodically. Cf. I-2 ; Proposition B]. The proof is given in Sect. 3. In Sect. 2 we develop the proof of the main calculation, modulo the analysis of periodic diffeomorphisms deferred to Sect. 3.

Groups associated to a knot in S3 which classify the knot have been constructed by Conway and Gordon [4], Simon [12] and Whitten[16]; see also [1] and [19] for certain special classes of knots. In this note we show that for a knot K in S3 of the groups π1(S3/K)/〈malb〉 classify the knot; here m resp. l denote a meridian resp. longitude of the knot and 〈〉 denotes normal closure. This is based on Thurston's orbifold geometrization theorem [13–15] and the following result (see also [8, 19]):Theorem 1. Let O1 and O2 be good closed irreducible 3-orbifolds which possess a decomposition into geometric pieces (along Euclidean 2-suborbifolds, see [2, 15]) and have infinite (orbifold-) fundamental group. Then O1 and O2 are diffeomorphic (as orbifolds) if and only if π1 O1 and π1 02 are isomorphic.

Publisher Summary This chapter presents an outline of Thurston's existence theorem for hyperbolic structures on three-dimensional manifolds-the three-dimensional uniformization theorem. It is based on lectures given by Thurston at the symposium on the Smith conjecture. The chapter provides a complete proof of the uniformization theorem. The criterion for deciding to treat the various pieces was simple: the parts of the argument that are more formal rely on the Bers–Ahlfors–Teichmuller theory or rely on three-dimensional topology. The statements of the main theorem concerning the existence of hyperbolic structures in the case when the manifolds have finite volume are considered. The chapter describes all the topological results necessary to carry out the inductive process of cutting apart the three-manifolds. The chapter focuses on Maskit's combination theorems thatdeal with gluing Kleinian groups together along common quasi-Fuchsian subgroups.