All Dehn Fillings of the Whitehead Link Complement are Tetrahedron Manifolds

Year 2022, , 192 - 201, 31.10.2022

Alberto Cavicchioli , Fulvia Spaggiari

https://doi.org/10.36890/iejg.1102753

Abstract

In this paper we show that Dehn surgeries on the oriented components of the Whitehead link yield tetrahedron manifolds of Heegaard genus $\le 2$. As a consequence, the eight homogeneous Thurston 3-geometries are realized by tetrahedron manifolds of Heegaard genus $\le 2$. The proof is based on techniques of Combinatorial Group Theory, and geometric presentations of manifold fundamental groups.

Keywords

Tetrahedron manifold, Whitehead link, Dehn surgery, Heegaard genus, group presentation

References

• [1] M. Aschenbrenner, S. Friedl and H. Wilton, Decision problems for 3-manifolds and their fundamental groups, Geometry & Topology Monographs 19 (2015), 201-236.
• [2] M. Brittenham and Y.Q. Wu, The classification of exceptional Dehnsurgeries on 2-bridge knots, Commun. Analysis Geom. 9 (1) (2001),97-113.
• [3] A. Cavicchioli, E. Molnár and F. Spaggiari, Some tetrahedron manifolds with Sol geometry and related groups, Journal of Geometry 105 (2014), 601-614.
• [4] A. Cavicchioli and F. Spaggiari, Tetrahedron manifold series of Heegaard genus two with knot presentation and Dehn surgery, Acta Math. Hungarica 131 (4) (2011), 307-322.
• [5] A. Cavicchioli, F. Spaggiari and A.I. Telloni, Topology of compact space forms from Platonic solids I, Topology and its Appl. 156 (2009), 812- 822.
• [6] A. Cavicchioli, F. Spaggiari and A.I. Telloni, Dehn surgeries on some classical links, Proceed. Edinburgh Math. Soc. 54 (2011), 33-45.
• [7] A.W.M. Dress, D.H. Huson and E. Molnár, The classification of face-transitive periodic three-dimensional tilings, Acta Crystallogr. A 49(1993), 806-817.
• [8] D.L. Johnson, Presentations of Groups, London Math. Soc. Stud. Texts, vol. 15, Cambridge Univ. Press, Cambridge, 1990.
• [9] R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer- Verlag, Berlin-Heidelberg-New York, 1976.
• [10] B. Martelli and C. Petronio, Dehn Filling of the "magic " 3-manifold, Commun. in Analysis and Geom. 14 (5) (2006), 969-1026.
• [11] A.D. Mednykh and A. Yu. Vesnin, On Heegaard genus of three-dimensional hyperbolic manifolds of small volume, Sb. Math. J. 37 (5) (1996), 893- 897.
• [12] A.D. Mednykh and A. Yu. Vesnin, Covering properties of small volume hyperbolic 3-manifolds, J. Knot Theory Ram. 7 (3) (1998), 381-392.
• [13] E. Molnár, Tetrahedron manifolds and space forms, Note Mat. 10 (1990), 335-346.
• [14] E. Molnár, Polyhedron complexes with simply transitive group actions and their realizations, Acta Math. Hung. 59 (1-2) (1992), 175-216.
• [15] E. Molnár, On non-Euclidean crystallography, some football manifolds, Structural Chemistry 23 (2012), 1057-1069.
• [16] E. Molnár and J. Szirmai, Symmetries in the 8 homogeneous 3-geometries, Symmetry: Culture and Science 21 (1-3) (2010), 87-117.
• [17] E. Molnár and J. Szirmai, Hyperbolic space forms with crystallographic applications and visualizations, In: International Conference on Geom- etry and Graphics, Springer Verlag, Cham (2018), pp. 320-337.
• [18] J. M. Montesinos, Classical Tesselations and Three-Manifolds, Universitext, Springer-Verlag, Berlin, 1987.
• [19] L. Moser, Elementary surgery along a torus knot, Pacific J. Math. 38(1971), 737-745.
• [20] G.D. Mostow, Strong rigidity of locally symmetric spaces, Princeton Univ. Press, Princeton, N.Y., 1973.
• [21] G. Prasad, Strong rigidity of Q-rank 1 lattices, Invent. Math. 21 (1973), 255-286.
• [22] P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-487.
• [23] J. Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), 88-111.
• [24] F. Spaggiari, On a theorem of L. Moser, Boll. U.M.I. 7-A (7) (1993), 421-429.
• [25] F. Spaggiari, The combinatorics of some tetrahedron manifolds, Discrete Math. 300 (2005), 163-179.