On trialities and their absolute geometries
-
Dimitri Leemans
,
Klara Stokes
Abstract
We introduce the notion of moving absolute geometry of a geometry with triality and show that, in the classical case where the triality is of type (Iσ) and the absolute geometry is a generalized hexagon, the moving 5 3 6 absolute geometry also gives interesting flag-transitive geometries with Buekenhout diagram 
for the groups G2(k) and 3D4(k), for any prime power k ≥ 2. We also classify the absolute geometries for geometries with trialities but no dualities coming from maps of Class III with automorphism group L2(q3), where q ≥ 2 is prime power. We then investigate the moving absolute geometries for these geometries, illustrating their interest in this case.
Funding statement: This research was made possible thanks to an Action de Recherche Concertée grant from the Communauté Française Wallonie-Bruxelles and the funding agency Knut and Alice Wallenberg Foundation grant numbers: KAW2020.0001, KAW2020.0007 and KAW 2020.0282.
Acknowledgements
The authors thank an anonymous referee whose comments helped to improve the paper.
References
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Articles in the same Issue
- Frontmatter
- Local quasi-isometries and tangent cones of definable germs
- On trialities and their absolute geometries
- An isomorphism between unitals and between related classical groups
- Betweenness isomorphisms in the plane — the case of a circle and points
- Compact locally homogeneous manifolds with parallel Weyl tensor
- Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving
- Equiangular lines in ℂ3
- Vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety
- Deformations of pairs of codimension one foliations
- Arithmetic counts of tropical plane curves and their properties
- Convex Fujita numbers are not determined by the fundamental group
Articles in the same Issue
- Frontmatter
- Local quasi-isometries and tangent cones of definable germs
- On trialities and their absolute geometries
- An isomorphism between unitals and between related classical groups
- Betweenness isomorphisms in the plane — the case of a circle and points
- Compact locally homogeneous manifolds with parallel Weyl tensor
- Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving
- Equiangular lines in ℂ3
- Vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety
- Deformations of pairs of codimension one foliations
- Arithmetic counts of tropical plane curves and their properties
- Convex Fujita numbers are not determined by the fundamental group
