Residual nilpotency of Fuchsian groups (Q793847)

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.