Residual nilpotency of Fuchsian groups
type: Item
instance of: scholarly article
title: Residual nilpotency of Fuchsian groups
zbMATH Open document ID: 0539.20026
author: A. Murray Macbeath
published in: Illinois Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 3857373
zbMATH Keywords: conjugacy classes of maximal finite cyclic subgroups
zbMATH Keywords: residually finite
zbMATH Keywords: residually nilpotent
zbMATH Keywords: cocompact Fuchsian groups
zbMATH Keywords: p-local groups
zbMATH Keywords: nilpotent automorphism groups of Riemann surfaces
reviewed by: C. Maclachlan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3857373