Mom technology and hyperbolic 3-manifolds

This is an expository paper on Mom-technology, describing the recent work of the authors in this area (found in arXiv:math/0606072, arXiv:0705.4325, and arXiv:0809.0346) concerning the use of Mom-technology to find the minimum-volume compact hyperbolic 3-manifold and the 10 smallest cusped hyperbolic 3-manifolds. In addition we provide a survey of a selection of results on volumes of hyperbolic 3-manifolds, and a discussion of outstanding open problems in this area.

[1]  R. Meyerhoff.  A Lower Bound for the Volume of Hyperbolic 3-Manifolds , 1987, Canadian Journal of Mathematics.

[2]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[3]  Armand Borel,et al.  Commensurability classes and volumes of hyperbolic 3-manifolds , 1981 .

[4]  C. Adams.  The noncompact hyperbolic 3-manifold of minimal volume , 1987 .

[5]  F. Gehring,et al.  Precisely invariant collars and the volume of hyperbolic 3-folds , 1998 .

[6]  Mathématiques DE L’I.H.É.S,et al.  Quasi-conformal mappings inn-space and the rigidity of hyperbolic space forms , 1968 .

[7]  Dehn filling, volume, and the Jones polynomial , 2006, math/0612138.

[8]  Sadayoshi Kojima,et al.  The smallest hyperbolic 3-manifolds with totally geodesic boundary , 1991 .

[9]  W. Thurston,et al.  Lower bounds on volumes of hyperbolic Haken 3-manifolds , 1999, math/0506338.

[10]  Cylinder and horoball packing in hyperbolic space , 2005 .

[11]  Volumes of Tubes in Hyperbolic 3-Manifolds , 2001 .

[12]  Ruth Kellerhals,et al.  On the volumes of hyperbolic 5-orthoschemes and the trilogarithm , 1992 .

[13]  David Gabai,et al.  MOM TECHNOLOGY AND VOLUMES OF HYPERBOLIC 3-MANIFOLDS , 2006, math/0606072.

[14]  Walter D. Neumann.  Hilbert’s 3rd Problem and Invariants of 3–manifolds , 1997 .

[15]  R. Meyerhoff.  The cusped hyperbolic 3-orbifold of minimum volume , 1985 .

[16]  Peter Milley.  Minimum volume hyperbolic 3‐manifolds , 2008, 0809.0346.

[17]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[18]  Andrew Przeworski.  A universal upper bound on density of tube packings in hyperbolic space , 2006 .

[19]  Harriet H. Moser.  Proving a manifold to be hyperbolic once it has been approximated to be so , 2008, 0809.1203.

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

[21]  J. Rubinstein,et al.  Existence of minimal surfaces of bounded topological type in three-manifolds , 1986 .

[22]  R. Benedetti,et al.  Lectures on Hyperbolic Geometry , 1992 .

[23]  M. Gromov.  Hyperbolic manifolds according to Thurston and Jørgensen , 1981 .

[24]  R. Meyerhoff.  Sphere-packing and volume in hyperbolic 3-space , 1986 .

[25]  W. Thurston.  The geometry and topology of 3-manifolds , 1979 .

[27]  Uffe Haagerup,et al.  Simplices of maximal volume in hyperbolicn-space , 1981 .

[28]  Über die dichteste Kugelpackung im hyperbolischen Raum , 1964 .

[29]  Yosuke Miyamoto,et al.  Volumes of hyperbolic manifolds with geodesic boundary , 1994 .

[30]  Universal bounds for hyperbolic Dehn surgery , 2002, math/0204345.

[31]  Gaven Martin,et al.  Minimal co-volume hyperbolic lattices, I: The spherical points of a Kleinian group , 2009 .

[32]  K. Böröczky.  Packing of spheres in spaces of constant curvature , 1978 .

[33]  J. Milnor.  Hyperbolic geometry: The first 150 years , 1982 .

[34]  M. Culler,et al.  Approximation by Maximal Cusps in Boundaries of Deformation Spaces of Kleinian Groups , 2003 .

[35]  Singular surfaces, mod 2 homology, and hyperbolic volume, I , 2005, math/0506396.

[36]  Dehn surgery, homology and hyperbolic volume , 2005, math/0508208.

[37]  The first betti number of the smallest closed hyperbolic 3-manifold , 1998 .

[38]  G. Robert Meyerhoff,et al.  The orientable cusped hyperbolic 3-manifolds of minimum volume , 2001 .

[39]  J. Przytycki,et al.  Hyperbolic structures on Dehn filling of some punctured-torus bundles over S1 , 1987 .

[40]  Abhijit Champanerkar,et al.  Exceptional Regions and Associated Exceptional Hyperbolic 3-Manifolds , 2007, Exp. Math..

[41]  David Gabai,et al.  Minimum volume cusped hyperbolic three-manifolds , 2007, 0705.4325.

[42]  J. Hass.  Acylindrical surfaces in $3$-manifolds. , 1995 .

[43]  Albert Marden,et al.  The Geometry of Finitely Generated Kleinian Groups , 1974 .

[44]  J. Ratcliffe.  Foundations of Hyperbolic Manifolds , 2019, Graduate Texts in Mathematics.

[45]  Gopal Prasad.  Strong rigidity ofQ-rank 1 lattices , 1973 .

[46]  Line arrangements in ℍ , 2005 .

[47]  JUN YANG,et al.  RATIONALITY PROBLEMS FOR K -THEORY AND CHERN-SIMONS INVARIANTS OF HYPERBOLIC 3-MANIFOLDS , 1995 .

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

[49]  F. Gehring,et al.  Inequalities for Möbius transformations and discrete groups. , 1991 .

[50]  I. Agol.  The minimal volume orientable hyperbolic 2-cusped 3-manifolds , 2008, 0804.0043.

[51]  C. Connell,et al.  Volume rigidity for finite volume manifolds , 2005 .