On démontre que, pour une large classe de groupes, le nombre de Reidemeister d'un automorphisme ϕ est égal au nombre de points fixes de dimension finie de sur le dual unitaire, si l'un de ces nombres est fini. Ce théorème est une généralisation naturelle aux groupes infinis du théorème classique de Burnside–Frobenius. Il a des conséquences importantes en dynamique topologique.
It is proved for a wide class of groups that the Reidemeister number of an automorphism ϕ is equal to the number of finite-dimensional fixed points of on the unitary dual, if one of these numbers is finite. This theorem is a natural generalization to infinite groups of the classical Burnside–Frobenius theorem. It has important consequences in Topological Dynamics.
Accepté le :
Publié le :
@article{CRMATH_2008__346_19-20_1033_0,
author = {Alexander Fel'shtyn and Evgenij Troitsky},
title = {Th\'eorie de {Burnside{\textendash}Frobenius} tordue pour les groupes virtuellement polycycliques},
journal = {Comptes Rendus. Math\'ematique},
pages = {1033--1038},
publisher = {Elsevier},
volume = {346},
number = {19-20},
year = {2008},
doi = {10.1016/j.crma.2008.09.003},
language = {fr},
}
TY - JOUR AU - Alexander Fel'shtyn AU - Evgenij Troitsky TI - Théorie de Burnside–Frobenius tordue pour les groupes virtuellement polycycliques JO - Comptes Rendus. Mathématique PY - 2008 SP - 1033 EP - 1038 VL - 346 IS - 19-20 PB - Elsevier DO - 10.1016/j.crma.2008.09.003 LA - fr ID - CRMATH_2008__346_19-20_1033_0 ER -
Alexander Fel'shtyn; Evgenij Troitsky. Théorie de Burnside–Frobenius tordue pour les groupes virtuellement polycycliques. Comptes Rendus. Mathématique, Volume 346 (2008) no. 19-20, pp. 1033-1038. doi : 10.1016/j.crma.2008.09.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.09.003/
[1] Dynamical zeta functions, Nielsen theory and Reidemeister torsion, Mem. Amer. Math. Soc., Volume 147 (2000) no. 699, p. xii+146
[2] The Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 279 (2001) no. 6, pp. 229-240 250 (Geom. i Topol.)
[3] Trace formulae, Zeta functions, congruences and Reidemeister torsion in Nielsen theory, Forum Math., Volume 10 (1998) no. 6, pp. 641-663
[4] Reidemeister numbers of Baumslag–Solitar groups, Algebra Discrete Math., Volume 3 (2006), pp. 36-48
[5] A twisted Burnside theorem for countable groups and Reidemeister numbers, Bonn, 2003 (K. Consani; M. Marcolli; Yu. Manin, eds.), Vieweg, Braunschweig (2006), pp. 141-154 (Preprint MPIM2004-65)
[6] Twisted Burnside theorem for type II1 groups: an example, Math. Res. Lett., Volume 13 (2006) no. 5, pp. 719-728
[7] Twisted conjugacy classes in exponential growth groups, Bull. London Math. Soc., Volume 35 (2003) no. 2, pp. 261-268
[8] Character Theory of Finite Groups, Academic Press, 1976
[9] On the automorphism group of generalised Baumslag–Solitar groups, 2005 (E-print arxiv:) | arXiv
[10] Most automorphisms of a hyperbolic group have very simple dynamics, Ann. Sci. École Norm. Sup., Volume 33 (2000), pp. 507-517
[11] Small cancellations over relatively hyperbolic groups and embedding theorems, 2004 (arxiv e-print) | arXiv
[12] Polycyclic Groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983
[13] Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003 (Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation)
[14] Noncommutative Riesz theorem and weak Burnside type theorem on twisted conjugacy, Funct. Anal. Appl., Volume 40 (2006) no. 2, pp. 117-125
Cité par Sources :
Commentaires - Politique
Étude d'une forme volume naturelle sur l'espace de représentations du groupe d'un nœud dans SU(2)
Jérôme Dubois
C. R. Math (2003)