Some remarks on proper actions, proper metric spaces, and buildings
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Linus Kramer
Abstract
We discuss various aspects of isometric group actions on proper metric spaces. As one application, we show that a proper and Weyl transitive action on a euclidean building is strongly transitive on the maximal atlas (the complete apartment system) of the building.
Funding statement: Funded by the Deutsche Forschungsgemeinschaft through a Polish-German Beethoven grant KR1668/11, and under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure.
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Communicated by: R. Weiss
Acknowledgements
Arthur Bartels’ request for a solid reference for Theorem 5.19 initiated a first, three page version of the present article. I thank Bertrand Rémy and Guy Rousseau for some remarks on Bruhat–Tits buildings. Stefan Witzel, Herbert Abels and Polychronis Strantzalos spotted mistakes and pointed out some more references. In particular, the authors kindly informed me that the forthcoming book [2] will contain the results that we present in Section 2. Richard Weiss made helpful comments on the proof of Theorem 5.19 and the material surrounding it. Philip Möller read the article, spotted several mistakes and made many good suggestions. I would also like to thank the referee, who raised several very good questions about an earlier version of the manuscript.
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Articles in the same Issue
- Frontmatter
- On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
-
A geographical study of
- A characterization of centrally symmetric convex bodies in terms of visual cones
- Closed Majorana representations of {3, 4}+-transposition groups
- Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
- Positive semigroups in lattices and totally real number fields
- A new axiomatics for masures II
- Helmut Salzmann and his legacy
- Some remarks on proper actions, proper metric spaces, and buildings
- Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case
- A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes
- Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35
