Frattini subgroups of $3$manifold groups
HTML articles powered by AMS MathViewer
 by R. B. J. T. Allenby, J. Boler, B. Evans, L. E. Moser and C. Y. Tang PDF
 Trans. Amer. Math. Soc. 247 (1979), 275300 Request permission
Abstract:
In this paper it is shown that if the Frattini subgroup of the fundamental group of a compact, orientable, irreducible, sufficiently large 3manifold is nontrivial then the 3manifold is a Seifert fibered space. We show further that the Frattini subgroup of the group of a Seifert fibered space is trivial or cyclic. As a corollary to our work we prove that every knot group has trivial Frattini subgroup.References

R. B. J. T. Allenby and C. Y. Tang, The Frattini subgroup of certain generalized free products and knot groups (preprint).
 R. B. J. T. Allenby and C. Y. Tang, On the Frattini subgroups of generalized free products, J. Algebra 52 (1978), no. 2, 460–474. MR 480759, DOI 10.1016/00218693(78)902508
 Gilbert Baumslag, On generalised free products, Math. Z. 78 (1962), 423–438. MR 140562, DOI 10.1007/BF01195185
 James W. Cannon and C. D. Feustel, Essential embeddings of annuli and Möbius bands in $3$manifolds, Trans. Amer. Math. Soc. 215 (1976), 219–239. MR 391094, DOI 10.1090/S00029947197603910941
 D. B. A. Epstein, Projective planes in $3$manifolds, Proc. London Math. Soc. (3) 11 (1961), 469–484. MR 152997, DOI 10.1112/plms/s311.1.469 B. Evans, The conjugacy problem for 3manifold groups, Ann. of Math. (submitted).
 Benny Evans and Louise Moser, Solvable fundamental groups of compact $3$manifolds, Trans. Amer. Math. Soc. 168 (1972), 189–210. MR 301742, DOI 10.1090/S00029947197203017426
 John Hempel, $3$Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
 John Hempel and William Jaco, Fundamental groups of $3$manifolds which are extensions, Ann. of Math. (2) 95 (1972), 86–98. MR 287550, DOI 10.2307/1970856
 Graham Higman and B. H. Neumann, On two questions of Itô, J. London Math. Soc. 29 (1954), 84–88. MR 57881, DOI 10.1112/jlms/s129.1.84
 A. Howard M. Hoare, Abraham Karrass, and Donald Solitar, Subgroups of infinite index in Fuchsian groups, Math. Z. 125 (1972), 59–69. MR 292948, DOI 10.1007/BF01111114
 William Jaco, Roots, relations and centralizers in threemanifold groups, Geometric topology (Proc. Conf., Park City, Utah, 1974) Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975, pp. 283–309. MR 0394668
 William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220
 Robert A. Katz and Wilhelm Magnus, Residual properties of free groups, Comm. Pure Appl. Math. 22 (1968), 1–13. MR 233873, DOI 10.1002/cpa.3160220102
 Ronnie Lee, Semicharacteristic classes, Topology 12 (1973), 183–199. MR 362367, DOI 10.1016/00409383(73)900062
 John Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957), 623–630. MR 90056, DOI 10.2307/2372566 K. Murasugi, Knot theory, Lecture notes, Univ. of Toronto, 1971.
 L. P. Neuwirth, Knot groups, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. MR 0176462
 L. P. Neuwirth, The status of some problems related to knot groups, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp. 209–230. MR 0431127
 Peter Orlik, Seifert manifolds, Lecture Notes in Mathematics, Vol. 291, SpringerVerlag, BerlinNew York, 1972. MR 0426001, DOI 10.1007/BFb0060329
 G. P. Scott, Compact submanifolds of $3$manifolds, J. London Math. Soc. (2) 7 (1973), 246–250. MR 326737, DOI 10.1112/jlms/s27.2.246
 W. R. Scott, Group theory, PrenticeHall, Inc., Englewood Cliffs, N.J., 1964. MR 0167513
 Jonathan Simon, Compactification of covering spaces of compact $3$manifolds, Michigan Math. J. 23 (1976), no. 3, 245–256 (1977). MR 431176
 C. Y. Tang, On the Frattini subgroups of generalized free products with cyclic amalgamations, Canad. Math. Bull. 15 (1972), 569–573. MR 314989, DOI 10.4153/CMB19720995
 Friedhelm Waldhausen, Gruppen mit Zentrum und $3$dimensionale Mannigfaltigkeiten, Topology 6 (1967), 505–517 (German). MR 236930, DOI 10.1016/00409383(67)900080
 Friedhelm Waldhausen, On irreducible $3$manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594 —, On the determination of some bounded 3manifolds by their fundamental groups alone, Proc. Internat. Sympos. Topology and Appl., HercegNovi, Yugoslavia, 1968: Beograd, 1969, pp. 331332.
 Alice Whittemore, On the Frattini subgroup, Trans. Amer. Math. Soc. 141 (1969), 323–333. MR 245687, DOI 10.1090/S00029947196902456879
Additional Information
 © Copyright 1979 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 247 (1979), 275300
 MSC: Primary 57M05; Secondary 20F34
 DOI: https://doi.org/10.1090/S00029947197905176958
 MathSciNet review: 517695