Malnormal subgroups and Frobenius groups: basics and examples
Confluentes Mathematici, Tome 6 (2014) no. 1, pp. 65-77.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/cml.13
Classification : 20B07, 20B05
Mots clés : Malnormal subgroup, infinite permutation group, Frobenius group, knot group, peripheral subgroup, almost nalmornal subgroup.
de la Harpe, Pierre 1 ; Weber, Claude 1

1 Section de mathématiques, Université de Genève, C.P. 64, CH–1211 Genève 4, Suisse
@article{CML_2014__6_1_65_0,
     author = {de la Harpe, Pierre and Weber, Claude},
     title = {Malnormal subgroups and {Frobenius} groups: basics and examples},
     journal = {Confluentes Mathematici},
     pages = {65--77},
     publisher = {Institut Camille Jordan},
     volume = {6},
     number = {1},
     year = {2014},
     doi = {10.5802/cml.13},
     mrnumber = {3266885},
     zbl = {1327.20030},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/cml.13/}
}
TY  - JOUR
AU  - de la Harpe, Pierre
AU  - Weber, Claude
TI  - Malnormal subgroups and Frobenius groups: basics and examples
JO  - Confluentes Mathematici
PY  - 2014
SP  - 65
EP  - 77
VL  - 6
IS  - 1
PB  - Institut Camille Jordan
UR  - http://www.numdam.org/articles/10.5802/cml.13/
DO  - 10.5802/cml.13
LA  - en
ID  - CML_2014__6_1_65_0
ER  - 
%0 Journal Article
%A de la Harpe, Pierre
%A Weber, Claude
%T Malnormal subgroups and Frobenius groups: basics and examples
%J Confluentes Mathematici
%D 2014
%P 65-77
%V 6
%N 1
%I Institut Camille Jordan
%U http://www.numdam.org/articles/10.5802/cml.13/
%R 10.5802/cml.13
%G en
%F CML_2014__6_1_65_0
de la Harpe, Pierre; Weber, Claude. Malnormal subgroups and Frobenius groups: basics and examples. Confluentes Mathematici, Tome 6 (2014) no. 1, pp. 65-77. doi : 10.5802/cml.13. http://www.numdam.org/articles/10.5802/cml.13/

[1] M. Aschbacher. Finite group theory, Second Edition, Cambridge Univ. Press, 2000. | DOI | Zbl

[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. | DOI | Zbl

[3] B. Baumslag. Generalized free products whose two-generator subgroups are free, J. Lond. Math. Soc., 43:601–606, 1968. | DOI | MR | Zbl

[4] G. Baumslag, A. Myasnikov and V. Remeslennikov. Malnormality is decidable in free groups, Int. J. Alg. Comput., 9(6):687–692, 1999. | DOI | MR | Zbl

[5] N. Bourbaki. Groupes et algèbres de Lie, chapitres 4, 5 et 6, Hermann, 1968. | Zbl

[6] M.R. Bridson and D.T. Wise. Malnormality is undecidable in hyperbolic groups, Isr. J. Math., 124:313–316, 2001. | DOI | MR | Zbl

[7] G. Burde and H. Zieschang. Knots, de Gruyter, 1985. | DOI

[8] M. Burger and S. Mozes. Lattices in products of trees, Publ. Math. IHÉS, 92:151–194, 2000. | DOI | Zbl

[9] R.G. Burns. A note on free groups, Proc. Amer. Math. Soc., 23:14–17, 1969. | DOI | MR | Zbl

[10] M.J. Collins. Some infinite Frobenius groups, J. Alg., 131(1):161–165, 1990. | DOI | MR | Zbl

[11] J.D. Dixon and B. Mortimer. Permutation groups, Springer, 1996. | DOI | Zbl

[12] B. Farb. Relatively hyperbolic groups, GAFA, 8(5):810–840, 1998. | DOI | MR | Zbl

[13] B. Fine, A. Myasnikov and G. Rosenberger. Malnormal subgroups of free groups, Comm. Alg., 20(9):4155–4164, 2002. | DOI | MR | Zbl

[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. | Zbl

[16] D. Gildenhuys, O. Kharlampovich and A. Myasnikov. CSA-groups and separated free constructions, Bull. Austr. Math. Soc., 52(1):63–84, 1995. | DOI | MR | Zbl

[17] M. Hall Jr. Coset representations in free groups, Trans. Amer. Math. Soc., 67:421–432, 1949. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[19] J. Hempel. 3–manifolds, Ann. Math. Stud., Princeton University Press, 1976. | Zbl

[20] B. Huppert. Endliche Gruppen I, Springer, 1967. | DOI | Zbl

[21] I.M. Isaacs. Finite group theory, Graduate Studies in Math. 92, Amer. Math. Soc., 2008. | DOI

[22] S.V. Ivanov. On some finiteness conditions in semigroup and group theory, Semigroup Forum, 48(1):28–36, 1994. | DOI | MR | Zbl

[23] I. Kapovich and A. Myasnikov. Stallings foldings and subgroups of free groups, J. Alg., 248(2):608–668, 2002. | DOI | MR | Zbl

[24] A. Karrass and D. Solitar. The free product of two groups with a malnormal amalgamated subgroup, Canad. J. Math., 23:933–959, 1971. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[27] O.H. Kegel and B.A.F. Wehrfritz. Locally finite groups, North-Holland, 1973. | DOI | Zbl

[28] W. Magnus, A. Karrass, and D. Solitar. Combinatorial group theory, Interscience, 1966. | DOI | Zbl

[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. | DOI | MR | Zbl

[30] P.M. Neumann and P.J. Rowley. Free actions of abelian groups on groups, 291–295, Lond. Math. Soc. Lec. Notes 252, 1998. | DOI | Zbl

[31] B.B. Newman. Some results on one-relator groups, Bull. Amer. Math. Soc., 74:568–571, 1968. | DOI | MR | Zbl

[32] D.V. Osin. Elementary subgroups of relatively hyperbolic groups and bounded generation, Int. J. Alg. Comput., 16(1):99–118, 2006. | DOI | MR | Zbl

[33] D.V. Osin. Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179, 843, 2006. | DOI | MR | Zbl

[34] D.V. Osin. Small cancellations over relatively hyperbolic groups and embedding theorems, Ann. Math., 172(1):1–39, 2010. | DOI | MR | Zbl

[35] J. Peterson and A. Thom. Group cocycles and the ring of affiliated operators, Inv. Math., 185(3):561–592, 2011. | DOI | MR | Zbl

[36] G. de Rham. Sur les polygones générateurs de groupes fuchsiens, L’Ens. Math., 17:49–61, 1971. | Zbl

[37] G. Robertson. Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces, J. Funct. Anal., 230(2):419–431, 2006. | DOI | MR | Zbl

[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. | DOI | Zbl

[39] J.R. Stallings. Topology of finite graphs, Inv. Math., 71:551–565, 1983. | DOI | MR | Zbl

[40] J.G. Thompson. Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. USA, 45:578–581, 1959. | DOI | MR | Zbl

[41] J.G. Thompson. Normal p-complements for finite groups, Math. Zeitschr., 72:332–354, 1960. | DOI | Zbl

[42] D.T. Wise. The residual finiteness of negatively curved polygons of finite groups, Inv. Math., 149(3):579–617, 2002. | DOI | MR | Zbl

[43] D.T. Wise. Residual finiteness of quasi-positive one-relator groups, J. Lond. Math. Soc. 66(2):334–350, 2002. | DOI | MR | Zbl

[44] D.T. Wise. A residually finite version of Rips’s construction, Bull. Lond. Math. Soc., 35(1):23–29, 2003. | DOI | MR | Zbl

[45] D.T. Wise. The structure of groups with a quasiconvex hierarchy, Electron. Res. Announc. Math. Sci., 16:44–55, 2009. | DOI | MR | Zbl

Cité par Sources :