Linear properties of poly-Fuchsian groups
type: Item
instance of: scholarly article
title: Linear properties of poly-Fuchsian groups
zbMATH Open document ID: 0824.20043
published in: Collectanea Mathematica
publication date: 1995-02-22T00:00:00Z
full work available at URL: https://eudml.org/doc/39673
review text: The poly-Fuchsian groups which are obtained by iterated extension of \(\text{PSL}_ 2(R)\)-lattices are usually not lattices in any semisimple group, and may display other interesting features. In this work the author shows the following results: (I) a poly-Fuchsian group contains a torsion free subgroup of finite index, (II) poly-Fuchsian groups are residually finite, (III) a solvable subgroup of a poly-Fuchsian group is polycyclic, whilst a non-solvable subgroup contains a non-abelian free subgroup, (IV) a torsion free poly-Fuchsian group is a duality group in the sense of Poincaré duality, (V) each poly-Fuchsian group contains a characteristic subgroup of finite index which is isomorphic to a direct product \(G_ 1 \times \dots \times G_ n\) where \(G_ 1, \dots, G_ n\) are irreducible poly-Fuchsian groups. Moreover, this decomposition is uniquely determined up to commensurability. (VI) If \(G\) is a poly- Fuchsian group with filtration \({\mathcal G} = (G_ r)_{0 \leq r \leq n}\), then the group \(\text{Aut}({\mathcal G})\) of filtration-preserving automorphisms is a subgroup of finite index in \(\Aut(G)\) and imbeds as a subgroup in \(\text{Aut} (Q_ 1) \times \dots \times \text{Aut}(Q_ n)\), where \(Q_ r = G_ r/G_{r - 1}\). In particular, \(\text{Aut} (G)\) is commensurable with a subgroup of \(\text{Aut} (Q_ 1) \times \dots \times \text{Aut} (Q_ n)\). The more geometrical aspects of poly-Fuchsian groups will be seen in the author's recent paper which is in preparation.
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 724157
zbMATH Keywords: semisimple groups
zbMATH Keywords: poly-Fuchsian groups
zbMATH Keywords: lattices
zbMATH Keywords: torsion free subgroup of finite index
zbMATH Keywords: residually finite groups
zbMATH Keywords: polycyclic groups
zbMATH Keywords: free subgroups
zbMATH Keywords: duality group
zbMATH Keywords: characteristic subgroup of finite index
zbMATH Keywords: irreducible poly-Fuchsian groups
zbMATH Keywords: decomposition
zbMATH Keywords: commensurability
zbMATH Keywords: filtration
zbMATH Keywords: filtration-preserving automorphisms
author: F. E. A. Johnson
reviewed by: Turgut Başkan
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 724157