A.Yu. Vesnin, T.A. Kozlovskaya. Brieskorn manifolds, generated Sieradski groups, and coverings of lens space ... С. 85-97

The Brieskorn manifold $\mathscr  B(p,q,r)$ is the $r$-fold cyclic covering of the three-dimensional sphere $S^{3}$ branched over the torus knot $T(p,q)$. The generalised Sieradski groups $S(m,p,q)$ are groups with $m$-cyclic presentation $G_{m}(w)$, where the word $w$ has a special form depending on $p$ and $q$. In particular, $S(m,3,2)=G_{m}(w)$ is the group with $m$ generators $x_{1},\ldots,x_{m}$ and $m$ defining relations $w(x_{i}, x_{i+1}, x_{i+2})=1$, where  $w(x_{i}, x_{i+1}, x_{i+2}) = x_{i} x_{i+2} x_{i+1}^{-1}$. Cyclic presentations of $S(2n,3,2)$ in the form  $G_{n}(w)$ were investigated by Howie and Williams, who showed that the $n$-cyclic presentations are geometric, i.e., correspond to the spines of closed three-dimensional manifolds. We establish an analogous result for the groups $S(2n,5,2)$. It is shown that in both cases the manifolds are $n$-fold branched cyclic coverings of lens spaces. For the classification of the constructed manifolds, we use Matveev's computer program "Recognizer".

Keywords: three-dimensional manifold, Brieskorn manifold, cyclically presented group, Sieradski group, lens space, branched covering.

The paper was received by the Editorial Office on August 7, 2017.

Andrei Yur’evich Vesnin, Dr. Phys.-Math. Sci, RAS Corresponding Member, Prof., Sobolev Institute of Mathematics, Novosibirsk, 630090 Russia; Novosibirsk State University, Novosibirsk, 630090 Russia,
e-mail: vesnin@math.nsc.ru

Tat’yana Anatol’evna Kozlovskaya, Cand. Sci. (Phys.-Math.), Magadan Institute of Economics, Magadan, 685000 Russia,
e-mail: konus_magadan@mail.ru

REFERENCES

1.   Brieskorn E. Beispiele zur Differentialtopologie von SingularitЈaten. Invent. Math., 1966, vol. 2, no. 1, pp. 1–14. doi: /10.1007/BF01403388 .

2.   Cavicchioli A., Hegenbarth F., Kim A. On cyclic branched coverings of torus knots. J. Geometry, 1999, vol. 64, pp. 55–66. doi: 10.1007/BF01229212 .

3.   Howie J., Williams G. Fibonacci type presentations and 3-manifolds, Topology Appl., 2017, vol. 215, pp. 24–34. doi: 10.1016/j.topol.2016.10.012 .

4.   Hempel J. 3-manifolds. Princeton, N.J.: Princeton University Press, 1976, Ser.  Annals of Math. Studies, vol. 86, 195 p. ISBN 978-0-8218-3695-8 .

5.   Matveev S. Algorithmic topology and classi cation of 3-manifolds, 2nd ed., Berlin: Springer, 2007, Ser. Algorithms Comput. Math., vol. 9, 492 pp. doi: 10.1007/978-3-540-45899-9 .

6.   Matveev S.V. Tabulation of three-dimensional manifolds. Russian Math. Surveys, 2005, vol. 60, no. 4, pp. 673–698. doi: 10.1070/RM2005v060n04ABEH003673 .

7.   Three-manifold Recognizer. The computer program developed by the research group of S. Matveev in the department of computer topology and algebra of Chelyabinsk State University.

8.   Weber C., Seifert H. Die Beiden DodekaederЈaume. Math. Z., 1933, vol. 37, pp. 237–253. doi: 10.1007/BF01474572 .

9.   Matveev S.V., Fomenko A.T. Constant energy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds. Russian Math. Surveys, 1988, vol. 43, no. 1, pp. 3–24. doi: 10.1070/RM1988v043n01ABEH001554 .

10.   Weeks J. Hyperbolic structures on 3-manifolds. Thesis (Ph.D.)–Princeton University, Princeton: Princeton University, 1985, 83 p.

11.   Mednykh A., Vesnin A. Visualization of the isometry group action on the Fomenko–Matveev–Weeks manifold. J. Lie Theory, 1998, vol. 8, no. 1, pp. 51–66.

12.   Helling H., Kim A., Mennicke J. A geometric study of Fibonacci groups. J. Lie Theory, 1998, vol. 8, no. 4, pp. 1–23.

13.   Sieradski A.J. Combinatorial squashings, 3-manifolds, and the third homology of groups. Invent. Math., 1986, vol. 84, pp.  121–139.

14.   Milnor J. Singular Points of Complex Hypersurfaces. Princeton: Princeton University Press and Tokyo University Press, 1968. 130 p. (Annals of Mathematics Studies). ISBN: 9781400881819 . Translated to Russian under the title Milnor Dzh. Osobye tochki kompleksnykh giperpoverkhnostei. Moscow, Mir Publ., 1971. 126 p.

15.   Milnor J. On the 3-dimensional Brieskorn manifolds M(p,q,r). In: Knots, Groups and 3-Manifolds. (Neuwirth L.P. Ed.), Ann. of Math. Studies, vol. 84, Princeton, N. J.: Princeton Univ. Press, 1975, pp. 175–225.

16.   Johnson D. Topics in the theory of group presentations, London Math. Soc. Lect. Note Ser., vol. 42, Cambridge: Cambridge Univ. Press, 1980. 320 p. ISBN: 978-0-521-23108-4 .

17.   Bardakov V.G., Vesnin A.Yu. A Generalization of Fibonacci Groups. Algebra and Logic, 2003, vol. 42, no. 2, pp. 73–91. doi: 10.1023/A:1023346206070 .

18.   Maclachlan C. Generalizations of Fibonacci numbers, groups and manifolds. In: Combinatorial and Geometric Group Theory (Duncan A.J., Gilbert N.D., Howie J. eds.), London Math. Soc. Lect. Note Ser., vol. 204, 1995, pp. 233–238. ISBN: 0521465958 .

19.   Johnson D.J., Wamsley J.W., Wright D. The Fibonacci groups. Proc. London Math. Soc., 1974, vol. 29, pp. 577–592. doi: 10.1112/plms/s3-29.4.577 .

20.   Szczepanski A. High dimensional knot groups and HNN extensions of the Fibonacci groups. J. Knot Theory Ramifications, 1998, vol. 7, pp. 503–508. doi: 10.1142/S0218216598000267 .

21.   Campbell C.M., Robertson E.F. A class of finitely presented groups of Fibonacci type. J. London Math. Soc., 1975, vol. 11, pp. 249–255. doi: 10.1112/jlms/s2-11.2.249 .

22.   Szczepanski A., Vesnin A. On generalized Fibonacci groups with odd number of generators. Communications in Algebra, 2000, vol. 28, no. 2, pp. 959–965. doi: 10.1080/00927870008826872 .

23.   Szczepanski A., Vesnin A. Generalized Neuwirth Groups and Seifert fibered manifolds, Algebra Colloquium, 2000, vol. 7, no. 3, pp. 295–303. doi: 10.1007/s10011-000-0295-7 .

24.   Neuwirth L. An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Camb. Philos. Soc., 1968, vol. 64, pp. 603–613. doi: 10.1017/S0305004100043279 .

25.   Johnson D.L., Mawdesley H. Some groups of Fibonacci type. J. Aust. Math. Soc., 1975, vol. 20, pp. 199–204. doi: 10.1017/S1446788700020498 .

26.   Gilbert N., Howie J. LOG groups and cyclically presented groups. J. Algebra, 1995, vol. 174, no. 1, pp. 118–131. doi: 10.1006/jabr.1995.1119 .

27.   Kim A.C., Vesnin A. Cyclically presented groups and Takahashi manifolds. Analysis of discrete groups, II (Kyoto, 1996), RIMS Kokyuroku, 1997, vol. 1022, pp. 200–212.

28.   Singer J. Three-dimensional manifolds and their Heegaard diagrams. Trans. Amer. Math. Soc., 1933, vol. 35, no. 1, pp. 88–111. doi: 10.1090/S0002-9947-1933-1501673-5 .

29.   Seifert H., Threlfall W. Lehrbuch Der Topologie. Chelsea Publishing, New York. 1934, 353 p. Translated from German to Russian under the title Zeifert G., Trel’fall’ V. Topologiya [Topology]. ONTI, 1938, 400 p.