Malnormal subgroups and Frobenius groups: basics and examples

Pierre de la Harpe[1]; Claude Weber[1]

  • [1] Section de mathématiques, Université de Genève, C.P. 64, CH–1211 Genève 4, Suisse

Confluentes Mathematici (2014)

  • Volume: 6, Issue: 1, page 65-76
  • ISSN: 1793-7434

Abstract

top
Malnormal subgroups occur in various contexts. We review a large number of examples, and compare the general situation to that of finite Frobenius groups of permutations.In a companion paper [18], we analyse when peripheral subgroups of knot groups and 3 -manifold groups are malnormal.

How to cite

top

de la Harpe, Pierre, and Weber, Claude. "Malnormal subgroups and Frobenius groups: basics and examples." Confluentes Mathematici 6.1 (2014): 65-76. <http://eudml.org/doc/275512>.

@article{delaHarpe2014,
abstract = {Malnormal subgroups occur in various contexts. We review a large number of examples, and compare the general situation to that of finite Frobenius groups of permutations.In a companion paper [18], we analyse when peripheral subgroups of knot groups and $3$-manifold groups are malnormal.},
affiliation = {Section de mathématiques, Université de Genève, C.P. 64, CH–1211 Genève 4, Suisse; Section de mathématiques, Université de Genève, C.P. 64, CH–1211 Genève 4, Suisse; Stevenson Center 1326, Department of Mathematics, Vanderbilt University Nashville, TN 37240, U.S.A.},
author = {de la Harpe, Pierre, Weber, Claude},
journal = {Confluentes Mathematici},
keywords = {Malnormal subgroup; infinite permutation group; Frobenius group; knot group; peripheral subgroup; almost nalmornal subgroup; malnormal subgroups; infinite permutation groups; finite Frobenius groups; knot groups; peripheral subgroups; almost malnormal subgroups},
language = {eng},
number = {1},
pages = {65-76},
publisher = {Institut Camille Jordan},
title = {Malnormal subgroups and Frobenius groups: basics and examples},
url = {http://eudml.org/doc/275512},
volume = {6},
year = {2014},
}

TY - JOUR
AU - de la Harpe, Pierre
AU - Weber, Claude
TI - Malnormal subgroups and Frobenius groups: basics and examples
JO - Confluentes Mathematici
PY - 2014
PB - Institut Camille Jordan
VL - 6
IS - 1
SP - 65
EP - 76
AB - Malnormal subgroups occur in various contexts. We review a large number of examples, and compare the general situation to that of finite Frobenius groups of permutations.In a companion paper [18], we analyse when peripheral subgroups of knot groups and $3$-manifold groups are malnormal.
LA - eng
KW - Malnormal subgroup; infinite permutation group; Frobenius group; knot group; peripheral subgroup; almost nalmornal subgroup; malnormal subgroups; infinite permutation groups; finite Frobenius groups; knot groups; peripheral subgroups; almost malnormal subgroups
UR - http://eudml.org/doc/275512
ER -

References

top
  1. M. Aschbacher. Finite group theory, Second Edition, Cambridge Univ. Press, 2000. Zbl0583.20001MR1777008
  2. H. Bass. Group actions on non-archimedean trees, in: Arboreal group theory, Proc. Workshop, Berkeley, 1988, 69–131, Publ. Math. Sci. Res. Inst. 19, 1991. Zbl0826.20026MR1105330
  3. B. Baumslag. Generalized free products whose two-generator subgroups are free, J. Lond. Math. Soc., 43:601–606, 1968. Zbl0172.02702MR233896
  4. G. Baumslag, A. Myasnikov and V. Remeslennikov. Malnormality is decidable in free groups, Int. J. Alg. Comput., 9(6):687–692, 1999. Zbl0949.20024MR1727165
  5. N. Bourbaki. Groupes et algèbres de Lie, chapitres 4, 5 et 6, Hermann, 1968. Zbl0186.33001MR240238
  6. M.R. Bridson and D.T. Wise. Malnormality is undecidable in hyperbolic groups, Isr. J. Math., 124:313–316, 2001. Zbl1018.20035MR1856523
  7. G. Burde and H. Zieschang. Knots, de Gruyter, 1985. Zbl1009.57003MR808776
  8. M. Burger and S. Mozes. Lattices in products of trees, Publ. Math. IHÉS, 92:151–194, 2000. Zbl1007.22013MR1839489
  9. R.G. Burns. A note on free groups, Proc. Amer. Math. Soc., 23:14–17, 1969. Zbl0184.04001MR252488
  10. M.J. Collins. Some infinite Frobenius groups, J. Alg., 131(1):161–165, 1990. Zbl0744.20003MR1055003
  11. J.D. Dixon and B. Mortimer. Permutation groups, Springer, 1996. Zbl0951.20001MR1409812
  12. B. Farb. Relatively hyperbolic groups, GAFA, 8(5):810–840, 1998. Zbl0985.20027MR1650094
  13. B. Fine, A. Myasnikov and G. Rosenberger. Malnormal subgroups of free groups, Comm. Alg., 20(9):4155–4164, 2002. Zbl1015.20021MR1936462
  14. F.G. Frobenius. Über auflösbare Gruppen IV, S’ber Akad. Wiss. Berlin, 1216–1230, 1901. [Gesammelte Abhandlungen III, 189–203, in particular page 196]. 
  15. E. Ghys and P. de la Harpe (éds). Sur les groupes hyperboliques d’après Mikhael Gromov, Birkhäuser, 1990. Zbl0731.20025MR1086648
  16. D. Gildenhuys, O. Kharlampovich and A. Myasnikov. CSA-groups and separated free constructions, Bull. Austr. Math. Soc., 52(1):63–84, 1995. Zbl0838.20025MR1344261
  17. M. Hall Jr. Coset representations in free groups, Trans. Amer. Math. Soc., 67:421–432, 1949. Zbl0035.01301MR32642
  18. P. de la Harpe and C. Weber. On malnormal peripheral subgroups of the fundamental group of a 3 -manifold, Confl. Math., 6:41–64, 2014. Zbl1319.57010
  19. J. Hempel. 3 –manifolds, Ann. Math. Stud., Princeton University Press, 1976. Zbl0345.57001MR415619
  20. B. Huppert. Endliche Gruppen I, Springer, 1967. Zbl0412.20002MR224703
  21. I.M. Isaacs. Finite group theory, Graduate Studies in Math. 92, Amer. Math. Soc., 2008. Zbl1169.20001MR2426855
  22. S.V. Ivanov. On some finiteness conditions in semigroup and group theory, Semigroup Forum, 48(1):28–36, 1994. Zbl0808.20046MR1245903
  23. I. Kapovich and A. Myasnikov. Stallings foldings and subgroups of free groups, J. Alg., 248(2):608–668, 2002. Zbl1001.20015MR1882114
  24. A. Karrass and D. Solitar. The free product of two groups with a malnormal amalgamated subgroup, Canad. J. Math., 23:933–959, 1971. Zbl0247.20028MR314992
  25. R. Kashaev. On ring-valued invariants of topological pairs, arXiv:math/07015432v2, 21 Jan 2007. 
  26. R. Kashaev. Δ -groupoids in knot theory, Geom. Dedicata, 150:105–130, 2011. Zbl1245.57015MR2753700
  27. O.H. Kegel and B.A.F. Wehrfritz. Locally finite groups, North-Holland, 1973. Zbl0259.20001MR470081
  28. W. Magnus, A. Karrass, and D. Solitar. Combinatorial group theory, Interscience, 1966. Zbl0138.25604
  29. A.G. Myasnikov and V.N. Remeslennikov. Exponential group 2: extensions of centralizers and tensor completion of CSA-groups, Int. J. Alg. Comput., 6(6):687–711, 1996. Zbl0866.20014MR1421886
  30. P.M. Neumann and P.J. Rowley. Free actions of abelian groups on groups, 291–295, Lond. Math. Soc. Lec. Notes 252, 1998. Zbl0952.20022MR1709963
  31. B.B. Newman. Some results on one-relator groups, Bull. Amer. Math. Soc., 74:568–571, 1968. Zbl0174.04603MR222152
  32. D.V. Osin. Elementary subgroups of relatively hyperbolic groups and bounded generation, Int. J. Alg. Comput., 16(1):99–118, 2006. Zbl1100.20033MR2217644
  33. D.V. Osin. Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179, 843, 2006. Zbl1093.20025MR2182268
  34. D.V. Osin. Small cancellations over relatively hyperbolic groups and embedding theorems, Ann. Math., 172(1):1–39, 2010. Zbl1203.20031MR2680416
  35. J. Peterson and A. Thom. Group cocycles and the ring of affiliated operators, Inv. Math., 185(3):561–592, 2011. Zbl1227.22003MR2827095
  36. G. de Rham. Sur les polygones générateurs de groupes fuchsiens, L’Ens. Math., 17:49–61, 1971. Zbl0214.28302MR335792
  37. G. Robertson. Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces, J. Funct. Anal., 230(2):419–431, 2006. Zbl1104.46031MR2186218
  38. G. Robertson and T. Steger. Malnormal subgroups of lattices and the Pukanszky invariant in group factors, J. Funct. Anal., 258(8):2708–2713, 2010. Zbl1204.22005MR2593340
  39. J.R. Stallings. Topology of finite graphs, Inv. Math., 71:551–565, 1983. Zbl0521.20013MR695906
  40. J.G. Thompson. Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. USA, 45:578–581, 1959. Zbl0086.25101MR104731
  41. J.G. Thompson. Normal p -complements for finite groups, Math. Zeitschr., 72:332–354, 1960. Zbl0098.02003MR117289
  42. D.T. Wise. The residual finiteness of negatively curved polygons of finite groups, Inv. Math., 149(3):579–617, 2002. Zbl1040.20024MR1923477
  43. D.T. Wise. Residual finiteness of quasi-positive one-relator groups, J. Lond. Math. Soc. 66(2):334–350, 2002. Zbl1049.20018MR1920406
  44. D.T. Wise. A residually finite version of Rips’s construction, Bull. Lond. Math. Soc., 35(1):23–29, 2003. Zbl1027.20014MR1934427
  45. D.T. Wise. The structure of groups with a quasiconvex hierarchy, Electron. Res. Announc. Math. Sci., 16:44–55, 2009. Zbl1183.20043MR2558631

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.