Linear properties of poly-Fuchsian groups
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.
scientific article; zbMATH DE number 724157