Group Extensions with Infinite Conjugacy Classes
Confluentes Mathematici, Tome 5 (2013) no. 1, pp. 73-92.

We characterize the group property of being with infinite conjugacy classes (or icc, i.e. infinite and of which all conjugacy classes except $\left\{1\right\}$ 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 $I{I}_{1}$, 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.

DOI : https://doi.org/10.5802/cml.3
Classification : 20E45,  20E22
@article{CML_2013__5_1_73_0,
author = {Pr\'eaux, Jean-Philippe},
title = {Group {Extensions} with {Infinite} {Conjugacy} {Classes}},
journal = {Confluentes Mathematici},
pages = {73--92},
publisher = {Institut Camille Jordan},
volume = {5},
number = {1},
year = {2013},
doi = {10.5802/cml.3},
mrnumber = {3143612},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/cml.3/}
}
TY  - JOUR
AU  - Préaux, Jean-Philippe
TI  - Group Extensions with Infinite Conjugacy Classes
JO  - Confluentes Mathematici
PY  - 2013
DA  - 2013///
SP  - 73
EP  - 92
VL  - 5
IS  - 1
PB  - Institut Camille Jordan
UR  - http://archive.numdam.org/articles/10.5802/cml.3/
UR  - https://www.ams.org/mathscinet-getitem?mr=3143612
UR  - https://doi.org/10.5802/cml.3
DO  - 10.5802/cml.3
LA  - en
ID  - CML_2013__5_1_73_0
ER  - 
Préaux, Jean-Philippe. Group Extensions with Infinite Conjugacy Classes. Confluentes Mathematici, Tome 5 (2013) no. 1, pp. 73-92. doi : 10.5802/cml.3. http://archive.numdam.org/articles/10.5802/cml.3/

[1] K. Brown, Cohomology of groups, Graduate Texts in Maths, 87, Springer-Verlag, 1982. | MR 672956 | Zbl 0584.20036

[2] M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes: les groupes hyperboliques de Gromov, Lecture Notes in Mathematics, 1441, Springer-Verlag, 1991. | MR 1075994 | Zbl 0727.20018

[3] Y. de Cornulier, Infinite conjugacy classes in groups acting on trees, Groups Geom. Dyn. 3(2):267–277, 2009. | MR 2486799 | Zbl 1186.20019

[4] J. Dixmier, von Neumann algebras, Translated from French by F. Jellett, Mathematical Library, 27, North-Holland, 1981. | MR 641217 | Zbl 0473.46040

[5] P. de la Harpe, On simplicity of reduced C*-algebras of groups, Bull. Lond. Math. Soc. 39:1–26, 2007. | MR 2303514 | Zbl 1123.22004

[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. | Numdam | MR 2379052 | Zbl 1213.57026

[7] R. Lyndon and P. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977. | MR 577064 | Zbl 0997.20037

[8] D. McDuff, Uncountably many $II-1$ factors, Ann. Math. 90(2):372–377, 1969. | MR 259625 | Zbl 0184.16902

[9] F.J. Murray and J. von Neumann, On rings of operators, IV, Ann. Math. 44:716–808, 1943. | MR 9096 | Zbl 0060.26903

[10] J. Rotman, An Introduction to the Theory of Groups, fourth edition, Graduate Texts in Mathematics, 148, Springer-Verlag, 1995. | MR 1307623 | Zbl 0810.20001

Cité par Sources :