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

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

2005-11-25

arxiv.org/abs/math/0511624v1

Publ. Math. Inst. Hautes \~A?tudes Sci. No. 104 (2006), 213-268.

Mathematics

Group Theory

Scientific paper

10.1007/s10240-006-0003-3

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(\Gamma, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.

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