Residual nilpotency of Fuchsian groups
A cocompact Fuchsian group \(\Gamma\) has a signature \((g;m_ 1,m_ 2,...,m_ r)\) which indicates, in particular, that \(\Gamma\) has r conjugacy classes of maximal finite cyclic subgroups and their orders are \(m_ 1,m_ 2,...,m_ r.\) All these groups are residually finite and except for the cases when \(g=0\) and the \(m_ i\) are coprime in pairs, in which case \(\Gamma\) is perfect, are known to be residually finite-and- soluble [\textit{C. H. Sah}, Acta. Math. 123, 13-42 (1969; Zbl 0208.100)]. The author proves that these Fuchsian groups are residually nilpotent if and only if the signature is p-local i.e. the \(m_ i\) are all powers of a single prime (this includes the torsion-free case). More generally, the study of homomorphisms from cocompact Fuchsian groups to nilpotent groups is reduced to the study of homomorphisms from p-local groups to finite p- groups. Using this, homomorphisms (with torsion free kernel) onto nilpotent groups are characterized. These are of particular interest as they give rise to nilpotent automorphism groups of Riemann surfaces.
conjugacy classes of maximal finite cyclic subgroups
nilpotent automorphism groups of Riemann surfaces
scientific article; zbMATH DE number 3857373