Discontinuous subgroups of PGL\(_{2}(K)\).
type: Item
instance of: scholarly article
title: Discontinuous subgroups of PGL\(_{2}(K)\).
zbMATH Open document ID: 1052.14029
DOI: 10.1016/S0021-8693(03)00398-3
author: M. van der Put
author: H. Voskuil
published in: Journal of Algebra
publication date: 2004-02-03T00:00:00Z
review text: Let \(K\) be a non-archimedean valued field of characteristic \(p\geq 0\) and with residue field \(k\) of characteristic \(p_k \geq 0\). The authors give a careful analysis of what are the possible finitely generated discontinuous subgroups \(\Gamma\) of \(PGL(2,K)\) that can act on \(\mathbb{P}^1(K)\) minus its set of limit points with quotient \(\mathbb{P}^1\). A free normal finite index subgroup of such a \(\Gamma\) gives rise to a finite branched cover of the projective line by a Mumford curve. The authors determine the possible structure of such \(\Gamma\) as amalgams of abstract finite groups and give a formula for the number of branch points in the corresponding cover (except if \(p=0\) and \(p_k \leq 5\)). This includes settling the question which (abstract) amalgams of groups are actually realizable as such \(\Gamma\) (compare: in \textit{G.~Cornelissen, F.~Kato} and \textit{A.~Kontogeorgis} [Math.\ Ann. 320, No. 1, 55--85 (2001; Zbl 1031.14011)] is a classification of abstract types of groups for \(p>0\) with \(\leq 3\) branch points, without addressing the realizability question).
reviewed by: Gunther Cornelissen
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11F23
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 2037091
zbMATH Keywords: non-archimedean valuation
zbMATH Keywords: Mumford curve
zbMATH Keywords: branched cover of the projective line
MaRDI profile type: Publication
cites work: Q4705677
cites work: Discontinuous groups in positive characteristic and automorphisms of Mumford curves
cites work: Géométrie analytique rigide et applications. (Ridgid analytic geometry and applications)
cites work: Schottky groups and Mumford curves
cites work: \(p\)-adic discontinuously embeddable graphs of groups
cites work: Mumford coverings of the projective line
cites work: Q3341041
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description: scientific article; zbMATH DE number 2037091