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Locally projective graphs of symplectic type

Alexander A. Ivanov

Vol. 20 (2023), No. 2-3, 303–315
Abstract

Locally projective graphs can be viewed as an axiomatised version of point-line graphs of the symplectic (thick) and orthogonal (thin) dual polar spaces over GF (2). Further examples come from Petersen and tilde geometries of the sporadic simple groups M22, M23, M24, He, Co2, Co1, J4, BM, M. The paper contributes to the classification of simply connected locally projective graphs. Such classification was accomplished for the thin case by S. V. Shpectorov and the author in 2002 with the essential use of results by V. I. Trofimov proved in a series of papers published in the 1990s. The graphs of the orthogonal dual polar spaces form the only nontrivial infinite series in the thin case. The orthogonal graphs are densely embedded in the symplectic graphs. The natural generalisation of this notion places in a general setting the famous embeddings

𝒢(M22)𝒢(M24),𝒢(Co2)𝒢(Co1),𝒢(BM)𝒢(M)

among Petersen and tilde geometries. It was recently proved by the author that graphs from a large class contain thin densely embedded subgraphs. In the present paper we characterise the symplectic dual polar graphs in the class of locally projective graphs with densely embedded subgraphs. The well-known special behaviour of the first cohomology group of the linear groups Ln(2) for n = 3 and 4 makes the proof a bit nontrivial but, unlike in the thin case, does not bring any exceptional examples.

To the memory of Jacques Tits

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Keywords
locally projective graph, dual polar space
Mathematical Subject Classification
Primary: 20D05, 20D06, 20D08
Milestones
Received: 28 July 2022
Revised: 4 May 2023
Accepted: 31 May 2023
Published: 13 September 2023
Authors
Alexander A. Ivanov
International Center for Mathematics
Southern University of Science and Technology
Shenzhen
China
Institute for Systems Analysis
Russian Academy of Sciences
FRC IAC RAS
Moscow
Russia