Automorphism groups of polycyclic-by-finite groups and arithmetic groups

Oliver Baues[1]; Fritz Grunewald

  • [1] ETH-Zürich, Departement Mathematik, Rämistrasse 101, Ch-8092 Zürich

Publications Mathématiques de l'IHÉS (2006)

  • Volume: 104, page 213-268
  • ISSN: 0073-8301

Abstract

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We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.

How to cite

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Baues, Oliver, and Grunewald, Fritz. "Automorphism groups of polycyclic-by-finite groups and arithmetic groups." Publications Mathématiques de l'IHÉS 104 (2006): 213-268. <http://eudml.org/doc/104220>.

@article{Baues2006,
abstract = {We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.},
affiliation = {ETH-Zürich, Departement Mathematik, Rämistrasse 101, Ch-8092 Zürich},
author = {Baues, Oliver, Grunewald, Fritz},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {polycyclic-by-finite groups; outer automorphism groups; arithmetic groups; rational homotopy theory},
language = {eng},
pages = {213-268},
publisher = {Springer},
title = {Automorphism groups of polycyclic-by-finite groups and arithmetic groups},
url = {http://eudml.org/doc/104220},
volume = {104},
year = {2006},
}

TY - JOUR
AU - Baues, Oliver
AU - Grunewald, Fritz
TI - Automorphism groups of polycyclic-by-finite groups and arithmetic groups
JO - Publications Mathématiques de l'IHÉS
PY - 2006
PB - Springer
VL - 104
SP - 213
EP - 268
AB - We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.
LA - eng
KW - polycyclic-by-finite groups; outer automorphism groups; arithmetic groups; rational homotopy theory
UR - http://eudml.org/doc/104220
ER -

References

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