On Seifert-fibre groups and the Bailey-Neumann theorem. (English) Zbl 1513.57009

Zomorrodian, Reza; Macbeath, Murray

JP J. Geom. Topol. 26, No. 1, 1-38 (2021).

Summary: In this paper, we investigate the Seifert-fiber groups further when they act as the fundamental groups of the 3-dimensional Seifert spaces. The groups under investigation are related to the fundamental groups in a similar way the Fuchsian groups are related to fundamental groups of Riemann surfaces [R. Zomorrodian, JP J. Geom. Topol. 9, No. 2, 169–188 (2009; Zbl 1258.20039), ibid. 14, No. 1, 87–98 (2013; Zbl 1298.20060) and ibid. 21, No. 2, 105–118 (2018; Zbl 1429.20034)]. We also discuss the invariant Euler number and its generalization as smooth circle bundle. The essential property of which has been discovered independently by Baily, Raymond Neumann, and Vasquez.

MSC:

57M07	Topological methods in group theory
57M05	Fundamental group, presentations, free differential calculus
20H10	Fuchsian groups and their generalizations (group-theoretic aspects)

Keywords:

Seifert fiber groups; Seifert fiber spaces; fundamental groups; Fuchsian groups; 3-dimensional Brieskorn manifolds; discrete subgroup; stabilizer class; Euler characteristic; Reidemeister-Schreier calculation

Citations:

Zbl 1258.20039; Zbl 1298.20060; Zbl 1429.20034

References:

[1]	Louis Auslander, F. Hahn and L. Green, Flows on Homogeneous Spaces, (AM-53), Volume 53, Princeton, 1963. · Zbl 0099.39103
[2]	G. O. Bailey, Uncharacteristically Euler!, Ph.D. Thesis, Birmingham, England, 1977.
[3]	P. E. Conner and F. Raymond, Actions of compact Lie groups on Aspherical manifolds, Topology of Manifolds (Proc. Inst., University of Georgia, Athens, Ga, 1969) and Markham, Chicago Ill, 1970, pp. 227-264. · Zbl 0312.57025
[4]	E. Brieskorn, Beispiele Zur Differential-Topologie Von Singularitaten, Inventiones, Math. 2 (1966), 1-14. · Zbl 0145.17804
[5]	I. V. Dolgacev, Automorphic forms and quasi-homogeneous singularities function, Anal. 9(2) (1975), 67-8. · Zbl 0321.14003
[6]	W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quarterly J. Math. (Oxford) 17(2) (1966), 86-97. · Zbl 0156.08901
[7]	A. H. M. H. Hoare, A. Karrass and D. Solitar, Subgroups of finite index of Fuchsian Groups, Math. Z. 120 (1971), 289-298. · Zbl 0223.20053
[8]	J. Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys 8, Amer. Math. Soc. Providence, R.I., 1964. · Zbl 0178.42902
[9]	R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Ergebnisse der Math., Vol. 89, Springer, Berlin, Heidelberg, New York, 1977. · Zbl 0368.20023
[10]	A. M. Macbeath, Discontinuous groups and birational transformations, Proc. Summer School, Queen’s College, Dundee, 1961 (Cyclostyled notes).
[11]	On Seifert-Fibre Groups and the Bailey-Neumann Theorem 37
[12]	A. M. Macbeath and D. Singerman, Spaces of subgroups and Teichmuller space, Proc. London Math. Soc. 31(3) (1975), 211-256. · Zbl 0314.32012
[13]	C. MacLachlan, Abelian Groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. 15(3) (1965), 699-712. · Zbl 0156.08902
[14]	B. Maskit, On Poincare’s theorem for fundamental polygons, Advances in Math. (1971), 219-230. · Zbl 0223.30008
[15]	M. H. Millington, Subgroups of the classical Modular groups, Journal of London Math. Soc. 1(2) (1969), 351-357. · Zbl 0206.36801
[16]	J. Milnor, On the 3-dimentional Brieskorn manifolds, P. 175-225 in “Knots Groups and 3-Manifolds”, Annals of Math. Studies 84, Princeton, 1975. · Zbl 0305.57003
[17]	J. Milnor and W. P. Thurston, On characteristic numbers for 3-manifolds, to appear. · Zbl 0393.57002
[18]	W. D. Neumann, 1 S actions and the -invariant of their involutions, Bonner Math. · Zbl 0219.57030
[19]	Schriften 44 (1970), Bonn. · JFM 57.1304.03
[20]	W. D. Neumann, Brieskorn complete intersections and automorphic forms, Inventions Math. 42 (1977), 285-293. · Zbl 0366.32015
[21]	W. D. Neumann and F. Raymond, Seifert Manifolds, Plumbing, μ -Invariant and orientation Reserving Maps. · Zbl 0401.57018
[22]	Peter Orlik, Seifert manifolds, Lecture Notes in Mathematics 291, Springer, Berlin, Heidelberg, New York, 1972. · Zbl 0263.57001
[23]	Peter Orlik, E. Vogt and H. Zieschang, Zur Topologie gefaserter dreidimentionaler Mannigfaltigkeiten, Topology 6 (1967), 49-64. · Zbl 0147.23503
[24]	R. C. Randell, The homology of generalized Brieskorn manifolds, Topology 14 (1975), 347-355. · Zbl 0317.57012
[25]	D. Singerman, Subgroups of Fuchsian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319-323. · Zbl 0206.30804
[26]	D. Singerman, On the structure of non-Euclidean crystallographic groups, Proc. Cambridge Phil. Soc. 76 (1974), 233-240. · Zbl 0284.20053
[27]	H. C. Wilkie, On non-Euclidean crystallographic groups, Math. Z. 91 (1966), 87-102. · Zbl 0166.02602
[28]	H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933), 147-238. · Zbl 0006.08304