Algorithmic detection and description of hyperbolic structures on closed 3{manifolds with solvable word problem

We outline a rigorous algorithm, rst suggested by Casson, for determining whether a closed orientable 3-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 1M .

[1]  A. Skinner.  The word problem in a class of non-haken 3-manifolds , 1994 .

[2]  Word hyperbolic Dehn surgery , 1998, math/9808120.

[3]  Edwin E. Moise,et al.  Affine structures in 3-manifolds, V, The triangulation theorem and Hauptvermutung , 1952 .

[4]  Friedhelm Waldhausen,et al.  Recent results on sufficiently large 3-manifolds , 1976 .

[5]  R. H. Bing,et al.  Locally Tame Sets are Tame , 1954 .

[6]  Graham A. Niblo,et al.  The geometry of cube complexes and the complexity of their fundamental groups , 1998 .

[7]  M. Taniguchi,et al.  Hyperbolic Manifolds and Kleinian Groups , 1998 .

[8]  A. Beardon.  The Geometry of Discrete Groups , 1995 .

[9]  Z. Sela,et al.  The isomorphism problem for hyperbolic groups I , 1995 .

[10]  Robert Riley,et al.  Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra , 1983 .

[11]  G. Robert Meyerhoff,et al.  Homotopy hyperbolic 3-manifolds are hyperbolic , 1996 .

[12]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[13]  William Jaco,et al.  Algorithms for the complete decomposition of a closed $3$-manifold , 1995 .

[14]  M. Gromov,et al.  Manifolds of Nonpositive Curvature , 1985 .

[15]  J. Weeks,et al.  Partially flat ideal triangulations of cusped hyperbolic 3-manifolds , 2000 .

[16]  J. Rubinstein,et al.  An Algorithm to Recognize the 3-Sphere , 1995 .

[17]  P. Scott,et al.  The geometries of 3-manifolds , 1983 .

[18]  Bruno Buchberger,et al.  Computer algebra symbolic and algebraic computation , 1982, SIGS.

[19]  John G. Ratcliffe,et al.  Geometry of Discrete Groups , 2019, Foundations of Hyperbolic Manifolds.

[20]  Friedhelm Waldhausen,et al.  The word problem in fundamental groups of sufficiently large irreducible 3-manifolds , 1968 .

[21]  B. Maskit,et al.  On Poincaré's theorem for fundamental polygons , 1971 .

[22]  Abigail Thompson,et al.  Thin Position and the Recognition Problem for $\bold{S^3}$ , 1994 .

[23]  Peter B. Shalen,et al.  Varieties of group representations and splittings of 3-manifolds , 1983 .

[24]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[25]  Geoffrey Hemion,et al.  On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds , 1979 .

[26]  S. D. Chatterji.  Proceedings of the International Congress of Mathematicians , 1995 .

[27]  Abigail Thompson,et al.  THIN POSITION AND THE RECOGNITION PROBLEM FOR S3 , 1994 .

[28]  Michelle Stocking,et al.  Almost normal surfaces in 3-manifolds , 1999 .

[29]  W. Thurston.  Three dimensional manifolds, Kleinian groups and hyperbolic geometry , 1982 .