Group Extensions with Infinite Conjugacy Classes

Jean-Philippe Préaux[1]

  • [1] Laboratoire d’Analyse, Topologie et Probabilités, UMR CNRS 7353, 39 rue F.Joliot-Curie, F-13453 Marseille cedex 13, France

Confluentes Mathematici (2013)

  • Volume: 5, Issue: 1, page 73-92
  • ISSN: 1793-7434

Abstract

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We characterize the group property of being with infinite conjugacy classes (or icc, i.e. infinite and of which all conjugacy classes except { 1 } are infinite) for groups which are extensions of groups. We prove a general result for extensions of groups, then deduce characterizations in semi-direct products, wreath products, finite extensions, among others examples we also deduce a characterization for amalgamated products and HNN extensions. The icc property is correlated to the Theory of von Neumann algebras since a necessary and sufficient condition for the von Neumann algebra of a discrete group Γ to be a factor of type I I 1 , is that Γ be icc. Our approach applies in full generality to the study of icc property since any group that does not split as an extension is simple, and in such case icc property becomes equivalent to being infinite.

How to cite

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Préaux, Jean-Philippe. "Group Extensions with Infinite Conjugacy Classes." Confluentes Mathematici 5.1 (2013): 73-92. <http://eudml.org/doc/275616>.

@article{Préaux2013,
abstract = {We characterize the group property of being with infinite conjugacy classes (or icc, i.e. infinite and of which all conjugacy classes except $\lbrace 1\rbrace $ are infinite) for groups which are extensions of groups. We prove a general result for extensions of groups, then deduce characterizations in semi-direct products, wreath products, finite extensions, among others examples we also deduce a characterization for amalgamated products and HNN extensions. The icc property is correlated to the Theory of von Neumann algebras since a necessary and sufficient condition for the von Neumann algebra of a discrete group $\Gamma $ to be a factor of type $II_1$, is that $\Gamma $ be icc. Our approach applies in full generality to the study of icc property since any group that does not split as an extension is simple, and in such case icc property becomes equivalent to being infinite.},
affiliation = {Laboratoire d’Analyse, Topologie et Probabilités, UMR CNRS 7353, 39 rue F.Joliot-Curie, F-13453 Marseille cedex 13, France},
author = {Préaux, Jean-Philippe},
journal = {Confluentes Mathematici},
keywords = {groups with infinite conjugacy classes; extensions of groups; semi-direct products; wreath products; finite extensions; amalgamated products; HNN extensions; split extensions},
language = {eng},
number = {1},
pages = {73-92},
publisher = {Institut Camille Jordan},
title = {Group Extensions with Infinite Conjugacy Classes},
url = {http://eudml.org/doc/275616},
volume = {5},
year = {2013},
}

TY - JOUR
AU - Préaux, Jean-Philippe
TI - Group Extensions with Infinite Conjugacy Classes
JO - Confluentes Mathematici
PY - 2013
PB - Institut Camille Jordan
VL - 5
IS - 1
SP - 73
EP - 92
AB - We characterize the group property of being with infinite conjugacy classes (or icc, i.e. infinite and of which all conjugacy classes except $\lbrace 1\rbrace $ are infinite) for groups which are extensions of groups. We prove a general result for extensions of groups, then deduce characterizations in semi-direct products, wreath products, finite extensions, among others examples we also deduce a characterization for amalgamated products and HNN extensions. The icc property is correlated to the Theory of von Neumann algebras since a necessary and sufficient condition for the von Neumann algebra of a discrete group $\Gamma $ to be a factor of type $II_1$, is that $\Gamma $ be icc. Our approach applies in full generality to the study of icc property since any group that does not split as an extension is simple, and in such case icc property becomes equivalent to being infinite.
LA - eng
KW - groups with infinite conjugacy classes; extensions of groups; semi-direct products; wreath products; finite extensions; amalgamated products; HNN extensions; split extensions
UR - http://eudml.org/doc/275616
ER -

References

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  6. P. de la Harpe and J.-P. Préaux, Groupes fondamentaux des variétés de dimension 3 et algèbres d’opérateurs, Ann. Fac. Sci. Toulouse Math., ser. 6, 16(3):561–589, 2007. Zbl1213.57026MR2379052
  7. R. Lyndon and P. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977. Zbl0997.20037MR577064
  8. D. McDuff, Uncountably many I I - 1 factors, Ann. Math. 90(2):372–377, 1969. Zbl0184.16902MR259625
  9. F.J. Murray and J. von Neumann, On rings of operators, IV, Ann. Math. 44:716–808, 1943. Zbl0060.26903MR9096
  10. J. Rotman, An Introduction to the Theory of Groups, fourth edition, Graduate Texts in Mathematics, 148, Springer-Verlag, 1995. Zbl0810.20001MR1307623

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