## Algorithmic detection and description of hyperbolic structures on closed 3-manifolds with solvable word problem (2001)

Citations: | 11 - 0 self |

### BibTeX

@MISC{Manning01algorithmicdetection,

author = {Jason Manning},

title = {Algorithmic detection and description of hyperbolic structures on closed 3-manifolds with solvable word problem},

year = {2001}

}

### OpenURL

### Abstract

We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable three-manifold M is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been given to solve the word problem in π1M.

### Citations

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Citation Context ...lating interval (or rectangle) with rational endpoints. A method for finding arbitrarily small isolating intervals for the solutions of a polynomial over Q can be found for instance in Section 6.2 of =-=[3]-=-. The method given there is for isolating real roots, 6but (as pointed out in an exercise) the method extends to complex roots as well. Since minimal polynomials are unique up to multiplication by an... |

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304 | Three-dimensional manifolds, Kleinian groups and hyperbolic geometry
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Citation Context ...d problem [7]. Lackenby [11] has shown that there are many three–manifolds with word hyperbolic fundamental group, not all of which are known to be hyperbolic. If Thurston’s geometrization conjecture =-=[26]-=- is true, then these manifolds are in fact hyperbolic. If M is any 3–manifold satisfying the geometrization conjecture, a solution to the word problem in its fundamental group may be found. Indeed, th... |

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Citation Context .../2 The norm ‖ · ‖ gives rise to a metric on the space of two–by–two complex matrices defined by δ(A,B) = ‖A−B‖. The restriction of this metric to SL2C gives the same topology as the ordinary one (see =-=[2]-=- for details). Therefore the hypothesis that G is indiscrete is equivalent to the existence of nontrivial elements A of G so that ‖A − I‖ is arbitrarily small. It is shown in [2] that if A ∈ SL2C acts... |

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Citation Context ... has solved the word problem in a class of non-Haken manifolds containing immersed surfaces of a particular type [23]. Manifolds with word hyperbolic fundamental group also have solvable word problem =-=[7]-=-. Lackenby [11] has shown that there are many three–manifolds with word hyperbolic fundamental group, not all of which are known to be hyperbolic. If Thurston’s geometrization conjecture [26] is true,... |

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Citation Context ...nifold satisfying the geometrization conjecture, a solution to the word problem in its fundamental group may be found. Indeed, the manifold may be first decomposed into prime pieces (see for instance =-=[10]-=-), whose fundamental groups form the free factors of π1M , and each of these pieces either has automatic fundamental group or is modeled on nilgeometry or solvgeometry (see [7], Chapter 12 for a proof... |

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Citation Context ...ery isometric sphere is a fundamental domain, sometimes called the Ford domain (though this term is sometimes used differently). In fact, this set is precisely the Dirichlet domain centered at 0 (see =-=[14]-=-, page 45). This is the fundamental domain which we will construct for a discrete action. According to Proposition 1.3 of [14], the isometric sphere of an isometry φ: B3 → B3 has Euclidean center φ−1 ... |

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Citation Context ...mental groups with solvable word problems; Waldhausen solved the word problem for Haken and virtually Haken manifolds [27], Niblo and Reeves for manifolds (of any dimension) with a nonnegative cubing =-=[15]-=-. Skinner has solved the word problem in a class of non-Haken manifolds containing immersed surfaces of a particular type [23]. Manifolds with word hyperbolic fundamental group also have solvable word... |

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Citation Context ...pace. Rubinstein and Rannard have claimed to be able to recognize small Seifert fibered spaces. Note that the homeomorphism problem for Haken 3-manifolds is decidable by work of Haken and Hemion [28],=-=[9]-=-. The homeomorphism problem is also decidable for “rigid weakly geometric” 3-manifolds (including Haken manifolds and all manifolds satisfying the geometrization conjecture) by the work of Sela [22]. ... |

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Citation Context ...ermine whether a 3–manifold is Haken is given in [10]. Although no known algorithm decides whether a 3–manifold is geometric, some particular geometric manifolds can be recognized algorithmically. In =-=[25]-=- Thompson gives an algorithm to recognize the 3–sphere (see also [18]). Similar methods provide recognition algorithms for the lens spaces [24]. Given the first homology of M (which is easily computed... |

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Citation Context ...Nutshell 2.1 Computations in Q The algorithms given in this paper rely heavily on the ability to do computations with algebraic numbers over the rationals. Most of what we need to use can be found in =-=[12]-=-. For the purposes of computation, each algebraic number is to be thought of as a minimal polynomial together with an isolating interval (or rectangle) with rational endpoints. A method for finding ar... |

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Citation Context ... input a representation ρ: π1M → SL2Q and decides if ρ is discrete and faithful with torsion free image. The plan of attack is to first try to build a fundamental region for the action, à la Riley in =-=[17]-=-, and then to use the solution to the word problem to verify that ρ is injective. It is convenient to consider both the ball and upper half–space models of hyperbolic space as subsets of the quaternio... |

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Citation Context ...ly constructible. 1.1 Context Several classes of manifolds are known to have fundamental groups with solvable word problems; Waldhausen solved the word problem for Haken and virtually Haken manifolds =-=[27]-=-, Niblo and Reeves for manifolds (of any dimension) with a nonnegative cubing [15]. Skinner has solved the word problem in a class of non-Haken manifolds containing immersed surfaces of a particular t... |

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Citation Context ...roperties are obvious. The fourth follows from and is made precise by the Margulis Lemma: Theorem 3.1 (Margulis Lemma) There exists a small positive constant µ .49 (e.g. µ = 16(2(4π/.49) 3 works, see =-=[1]-=-, page 107) such that the sugbroup +1) Γµ(M,x) ⊂ π1(M,x) generated by loops based at x ∈ M of length at most µ is abelian, for any hyperbolic three–manifold M , and any x ∈ M . In particular, if M is ... |

9 |
Locally tame sets are tame
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Citation Context ...he homeomorphism f: M → N , we say that the geometric structure on M is algorithmically constructible. Remark 1.2 If two triangulated compact 3–manifolds M and N are homeomorphic, then it is shown in =-=[4]-=- that the given triangulations admit isomorphic subdivisions. For any n, there are finitely many subdivisions of the triangulations of N and M with n tetrahedra, so we can find them all and test each ... |

7 |
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Citation Context ...lation of M we obtain some presentation of π1M : π1M = 〈g1,... ,gn|w1,... ,wm〉 (1) We then consider the representation variety of homomorphisms R = {ρ: π1M → SL2C} ⊆ C 4n 7as defined for example, in =-=[6]-=-. Since π1M is finitely presented, this variety is constructible as the zero set of a finite set of polynomials with integer coefficients. Indeed, SL2C can be thought of as the variety in C 4 defined ... |

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3 |
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Citation Context ...blo and Reeves for manifolds (of any dimension) with a nonnegative cubing [15]. Skinner has solved the word problem in a class of non-Haken manifolds containing immersed surfaces of a particular type =-=[23]-=-. Manifolds with word hyperbolic fundamental group also have solvable word problem [7]. Lackenby [11] has shown that there are many three–manifolds with word hyperbolic fundamental group, not all of w... |

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2 |
Partially flat ideal triangulations of cusped 3{manifolds, Osaka
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Citation Context ...yperbolic structures on the remaining pieces can be computed by a variety of methods, for instance by a variation on the methods of the present paper. Alternatively, results of and Petronio and Weeks =-=[16]-=- have shown that a manifold with torus boundary components admits a finite volume hyperbolic structure if and only if Thurston’s hyperbolization equations for some ideal triangulation have a solution ... |

2 |
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Citation Context ...those face identifications is answered by the Poincaré Polyhedron Theorem. We use the theorem in the following specialized form (more general versions can be found in the standard references [13] and =-=[20]-=-): Theorem 4.8 (Poincaré Polyhedron Theorem) Let D be a finite sided polyhedron in hyperbolic space with no ideal vertices, together with face pairings which are orientation-preserving isometries gene... |

2 |
Almost normal surfaces
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Citation Context ...eometric manifolds can be recognized algorithmically. In [25] Thompson gives an algorithm to recognize the 3–sphere (see also [18]). Similar methods provide recognition algorithms for the lens spaces =-=[24]-=-. Given the first homology of M (which is easily computed) there is a finite list of lens spaces 3which M could be homeomorphic to, so we can determine whether a manifold is a lens space. Rubinstein ... |

2 | Ane structures in 3{manifolds,V: the triangulation theorem - Moise - 1952 |

1 |
results on sufficiently large 3–manifolds, Algebraic and geometric topology
- Recent
- 1978
(Show Context)
Citation Context ...ens space. Rubinstein and Rannard have claimed to be able to recognize small Seifert fibered spaces. Note that the homeomorphism problem for Haken 3-manifolds is decidable by work of Haken and Hemion =-=[28]-=-,[9]. The homeomorphism problem is also decidable for “rigid weakly geometric” 3-manifolds (including Haken manifolds and all manifolds satisfying the geometrization conjecture) by the work of Sela [2... |