Vol. 17, No. 3, 2019

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On two nonbuilding but simply connected compact Tits geometries of type $C_3$

Antonio Pasini

Vol. 17 (2019), No. 3, 221–249
Abstract

A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group acting continuously on the geometry, has been obtained by Kramer and Lytchak (2014; 2019). According to their main result, all such geometries but two are quotients of buildings. The two exceptions are flat geometries of type C3 and arise from polar actions on the Cayley plane over the division algebra of real octonions. The classification obtained by Kramer and Lytchak does not contain the claim that those two exceptional geometries are simply connected, but this holds true, as proved by Schillewaert and Struyve (2017). Their proof is of topological nature and relies on the main result of (Kramer and Lytchak 2014; 2019). In this paper we provide a combinatorial proof of that claim, independent of (Kramer and Lytchak 2014; 2019).

Keywords
compact geometries, composition algebras, diagram geometries
Mathematical Subject Classification 2010
Primary: 20E42, 51E24, 57S15
Milestones
Received: 27 November 2018
Revised: 12 March 2019
Accepted: 27 April 2019
Published: 9 October 2019
Authors
Antonio Pasini
Department of Information Engineering and Mathematics
University of Siena
Siena
Italy