\(4\)-punctured spheres
type: Item
instance of: scholarly article
title: \(4\)-punctured spheres
zbMATH Open document ID: 0829.20071
author: John G. Thompson
published in: Journal of Algebra
publication date: 1996-01-15T00:00:00Z
review text: If \((g;e_1, \dots, e_r; m)\) is the signature of a Fuchsian group of the first kind, let \({\mathcal F}(g; e_1, \dots, e_r; m)\) denote the set of all Fuchsian groups of that signature. The author obtains very precise information on the set \({\mathcal F}(0; -; 4)\) i.e. the monodromy groups of (hyperbolic) 4-punctured spheres. More precisely, if \(I(G)\) denotes the subgroup generated by all involutions in \(G\), and \(G^+\), the normaliser in \(\text{PSL} (2, \mathbb{R})\), then the map \(G \to I(G^+)\) is a bijection from \({\mathcal F} (0; -; 4)\) to \({\mathcal F} (0;2,2,2;1)\) and three distinct possibilities for \(G\) arise, determined by the nature of \(G^+/G\), being a 4-group, a dihedral group of order 8, or isomorphic to \(\text{PSL} (2,3)\). The methods are direct, but cunning calculations. The braid group on 4- strings is related to the mapping class group of the 4-punctured sphere, and some of the calculations are shown to be motivated by an invariance under certain braid group actions.
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 735318
zbMATH Keywords: hyperbolic 4-punctured spheres
zbMATH Keywords: Fuchsian groups
zbMATH Keywords: monodromy groups
zbMATH Keywords: involutions
zbMATH Keywords: mapping class groups
zbMATH Keywords: braid group actions
reviewed by: C. Maclachlan
MaRDI profile type: Publication
full work available at URL: jabr.1995.1029
OpenAlex ID: W1972291283
DOI: 10.1006/JABR.1995.1029
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description: scientific article; zbMATH DE number 735318