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.
@article{PMIHES_2006__104__213_0,
author = {Baues, Oliver and Grunewald, Fritz},
title = {Automorphism groups of polycyclic-by-finite groups and arithmetic groups},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {213--268},
publisher = {Springer},
volume = {104},
year = {2006},
doi = {10.1007/s10240-006-0003-3},
mrnumber = {2264837},
zbl = {1121.20027},
language = {en},
url = {http://archive.numdam.org/articles/10.1007/s10240-006-0003-3/}
}
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 SP - 213 EP - 268 VL - 104 PB - Springer UR - http://archive.numdam.org/articles/10.1007/s10240-006-0003-3/ DO - 10.1007/s10240-006-0003-3 LA - en ID - PMIHES_2006__104__213_0 ER -
%0 Journal Article %A Baues, Oliver %A Grunewald, Fritz %T Automorphism groups of polycyclic-by-finite groups and arithmetic groups %J Publications Mathématiques de l'IHÉS %D 2006 %P 213-268 %V 104 %I Springer %U http://archive.numdam.org/articles/10.1007/s10240-006-0003-3/ %R 10.1007/s10240-006-0003-3 %G en %F PMIHES_2006__104__213_0
Baues, Oliver; Grunewald, Fritz. Automorphism groups of polycyclic-by-finite groups and arithmetic groups. Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 213-268. doi : 10.1007/s10240-006-0003-3. http://archive.numdam.org/articles/10.1007/s10240-006-0003-3/
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