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Some series of honey-comb spaces. (English) Zbl 1186.57002

Starting with a detailed discussion of the Seifert-Weber dodecahedral space the authors study a family \(M_{n,k,l}\) of closed orientable \(3\)-manifolds obtained as a quotient space of a polyhedron with certain edge identifications and introduced by A. C. Kim, Y. Kim and S. H. Kim [A. C. Kim and Y. Kim, Res. Expo. Math. 27, 65–73 (2003; Zbl 1058.57002); S. H. Kim and Y. Kim, Commun. Korean Math. Soc. 20, No. 4, 803–812 (2005; Zbl 1083.57003)]. They obtain geometric presentations of their fundamental groups and their split extension groups and determine which of the \(M_{n,k,l}\) are hyperbolic and which are Seifert fibered. The cyclic covers of \(M_{n,k,l}\) over \(S^3\) branched over well-known links are described. Finally the authors obtain a classification of a certain class of honey-comb spaces constructed by A. C. Kim and A. I. Kostrikin [Dokl. Math. 51, No. 1, 36–38 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 340, No. 2, 158–160 (1995; Zbl 0907.57001); Sb. Math. 188, No.2, 173-194, Errata No.9, 1415 (1997); translation from Mat. Sb. 188, No. 2, 3-24 (1997); errata ibid. 188, No. 9, 157 (1997; Zbl 0906.20021)].

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57M60 Group actions on manifolds and cell complexes in low dimensions
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[1] E. Barbieri, A. Cavicchioli and F. Spaggiari, Seifert hyperelliptic manifolds , Intern. J. Pure Appl. Math. 6 (2003), 317-342. · Zbl 1054.57002
[2] G. Burde and H. Zieschang, Knots , Walter de Gruyter, Berlin, 1985.
[3] A. Cavicchioli and L. Paoluzzi, On certain classes of hyperbolic \(3\)-manifolds , Manuscripta Math. 101 (2000), 457-494. · Zbl 0947.57013 · doi:10.1007/s002290050227
[4] A. Cavicchioli, B. Ruini and F. Spaggiari, Cyclic branched coverings of \(2\)-bridge knots , Revista Matematica Univ. Complutense Madrid 12 (1999), 383-416. · Zbl 0952.57001
[5] H.S.M. Coxeter and W.O. Moser, Generators and relations for discrete groups , Springer Verlag, Berlin, 1957. · Zbl 0077.02801
[6] F. Grunewald and U. Hirsch, Link complements arising from arithmetic group actions , International J. Math. 6 (1995), 337-370. · Zbl 0840.57007 · doi:10.1142/S0129167X95000109
[7] H. Helling, A.C. Kim and J.L. Mennicke, Some honey-combs in hyperbolic \(3\)-space , Comm. Algebra 23 (1995), 5169-5206. · Zbl 1002.57503 · doi:10.1080/00927879508825526
[8] S.R. Henry and J.R. Weeks, Symmetry groups of hyperbolic knots and links , J. Knot Theory Ramifications 1 (1992), 185-201. · Zbl 0757.57008 · doi:10.1142/S0218216592000100
[9] H.M. Hilden, M.T. Lozano and J.M. Montesinos, On the arithmetic \(2\)-bridge knots and link orbifolds and a new knot invariant , J. Knot Theory Ramifications 4 (1995), 81-114. · Zbl 0844.57006 · doi:10.1142/S0218216595000053
[10] A. Kawauchi, A Survey of knot theory , Birkhäuser Verlag, Basel, 1996. · Zbl 0861.57001
[11] A.C. Kim and Y. Kim, On generalized Whitehead links and \(3\)-manifolds , in Recent advances in group theory and low-dimensional topology , · Zbl 1058.57002
[12] A.C. Kim and A.I. Kostrikin, Three series of \(3\)-manifolds and their fundamental groups , Dokl. Akad. Nauk. 340 (1995), 158-160. · Zbl 0907.57001
[13] ——–, Certain balanced groups and \(3\)-manifolds , Mat. Sbor. 188 (1997), 3-24 (in Russian); Sbor. Math. 188 (1997), 173-194 (in English). · Zbl 0906.20021 · doi:10.1070/SM1997v188n02ABEH000193
[14] S.H. Kim and Y. Kim, On certain classes of links and \(3\)-manifolds , Comm. Korean Math. Soc. 20 (2005), 803-812. · Zbl 1083.57003 · doi:10.4134/CKMS.2005.20.4.803
[15] Y. Kim and A. Vesnin, On the Johnson’s question about the Kim-Kostrikin group , Comm. Korean Math. Soc. 11 (1996), 933-936. · Zbl 0946.20012
[16] P. Lorimer, Four dodecahedral spaces , Pacific J. Math. 156 (1992), 329-335. · Zbl 0770.57011 · doi:10.2140/pjm.1992.156.329
[17] A.D. Mednykh and A. Vesnin, On the volume of hyperbolic Whitehead link cone-manifolds , Geometry Anal. Sci. Math. Sci. 8 (2002), 1-11. · Zbl 1104.57300
[18] J. Minkus, The branched cyclic coverings of \(2\)-bridge knots and links , Memoirs Amer. Math. Soc. 255 , Providence, RI, 1982. · Zbl 0491.57005
[19] M. Mulazzani and A. Vesnin, The many faces of cyclic branched coverings of \(2\)-bridge knots and links , Atti Sem. Mat. Fis. Univ. Modena, Suppl. 49 · Zbl 1221.57009
[20] D. Rolfsen, Knots and links , Math. Lecture Series 7 , Publish or Perish Inc., Berkeley, 1976. · Zbl 0339.55004
[21] M. Sakuma, The geometries of spherical Montesinos links , Kobe J. Math. 7 (1990), 167-190. · Zbl 0727.57007
[22] H. Seifert and W. Threlfall, A textbook of topology , Academic Press, New York, 1980. · Zbl 0469.55001
[23] W.P. Thurston, The geometry and topology of \(3\)-manifolds , Lecture Notes, Princeton University Press, Princeton, NJ, 1980.
[24] B. Zimmermann, On cyclic branched coverings of hyperbolic links , Topology Appl. 65 (1995), 287-294. · Zbl 0848.57012 · doi:10.1016/0166-8641(95)00007-4
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