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Amalgams Determined By Locally Projective Actions

Published online by Cambridge University Press:  22 January 2016

A. A. Ivanov  and
S. V. Shpectorov
A. A. Ivanov
Affiliation:
Department of Mathematics, Imperial College, 180 Queen’s Gate, London, SW7 2BZ, UK
S. V. Shpectorov
Affiliation:
Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, USA
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Abstract

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A locally projective amalgam is formed by the stabilizer G(x) of a vertex x and the global stabilizer G{x, y} of an edge (containing x) in a group G, acting faithfully and locally finitely on a connected graph Γ of valency 2n - 1 so that (i) the action is 2-arc-transitive; (ii) the subconstituent G(x)Γ(x) is the linear group SLn(2) = Ln(2) in its natural doubly transitive action and (iii) [t, G{x, y}] < O2(G(x) n G{x, y}) for some t G G{x, y} \ G(x). D. Z. Djokovic and G. L. Miller [DM80], used the classical Tutte’s theorem [Tu47], to show that there are seven locally projective amalgams for n = 2. Here we use the most difficult and interesting case of Trofimov’s theorem [Tr01] to extend the classification to the case n > 3. We show that besides two infinite series of locally projective amalgams (embedded into the groups AGLn(2) and O2n+(2)) there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M22, M23, Co2, J4 and BM. For each of the exceptional amalgam n = 3, 4 or 5.

Keywords

20B2520E0820D08
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 176 , 2004 , pp. 19 - 98
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

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