Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial

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

Antonio Pasini

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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).

Article information

Innov. Incidence Geom. Algebr. Topol. Comb., Volume 17, Number 3 (2019), 221-249.

Received: 27 November 2018
Revised: 12 March 2019
Accepted: 27 April 2019
First available in Project Euclid: 29 October 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 51E24: Buildings and the geometry of diagrams 57S15: Compact Lie groups of differentiable transformations

compact geometries composition algebras diagram geometries


Pasini, Antonio. On two nonbuilding but simply connected compact Tits geometries of type $C_3$. Innov. Incidence Geom. Algebr. Topol. Comb. 17 (2019), no. 3, 221--249. doi:10.2140/iig.2019.17.221.

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