Isometries of wall-connected twin buildings
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Sebastian Bischof
Abstract
We introduce the notion of a wall-connected twin building and show that the local-to-global principle holds for these twin buildings. As each twin building satisfying Condition (co) (introduced in [7]) is wall-connected, we obtain a strengthening of the main result of [7] that covers also the thick irreducible affine twin buildings of rank at least 3.
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Communicated by: R. Weiss
Acknowledgements
We thank Richard Weiss for communicating to us the proof of Proposition 6.13. Moreover, we thank the anonymous referee for his/her detailed comments and helpful suggestions.
References
[1] P. Abramenko, K. S. Brown, Buildings. Springer 2008. MR2439729 Zbl 1214.2003310.1007/978-0-387-78835-7Search in Google Scholar
[2] S. Bischof, A. Chosson, B. Mühlherr, On isometries of twin buildings. Beitr. Algebra Geom. 62 (2021), 441–456. MR4254628 Zbl 1482.5100610.1007/s13366-020-00524-5Search in Google Scholar
[3] N. Bourbaki, Lie groups and Lie algebras. Chapters 4–6. Springer 2002. MR1890629 Zbl 0983.1700110.1007/978-3-540-89394-3Search in Google Scholar
[4] A. Devillers, B. Mühlherr, H. Van Maldeghem, Codistances of 3-spherical buildings. Math. Ann. 354 (2012), 297–329. MR2957628 Zbl 1259.5100510.1007/s00208-011-0733-5Search in Google Scholar
[5] A. W. M. Dress, R. Scharlau, Gated sets in metric spaces. Aequationes Math. 34 (1987), 112–120. MR915878 Zbl 0696.5402210.1007/BF01840131Search in Google Scholar
[6] B. Mühlherr, H. P. Petersson, R. M. Weiss, Descent in buildings, volume 190 of Annals of Mathematics Studies. Princeton Univ. Press 2015. MR3364836 Zbl 1338.5100210.1515/9781400874019Search in Google Scholar
[7] B. Mühlherr, M. Ronan, Local to global structure in twin buildings. Invent. Math. 122 (1995), 71–81. MR1354954 Zbl 0854.5100910.1007/BF01231439Search in Google Scholar
[8] B. Mühlherr, H. Van Maldeghem, On certain twin buildings over tree diagrams. Bull. Belg. Math. Soc. Simon Stevin 5 (1998), 393–402. MR1630040 Zbl 0934.5100810.36045/bbms/1103409019Search in Google Scholar
[9] M. A. Ronan, Local isometries of twin buildings. Math. Z. 234 (2000), 435–455. MR1774092 Zbl 1128.2030710.1007/PL00004808Search in Google Scholar
[10] J. Tits, Buildings of spherical type and finite BN-pairs. Springer 1974. MR0470099 Zbl 0295.20047Search in Google Scholar
[11] J. Tits, Uniqueness and presentation of Kac-Moody groups over fields. J. Algebra 105 (1987), 542–573. MR873684 Zbl 0626.2201310.1016/0021-8693(87)90214-6Search in Google Scholar
[12] J. Tits, Twin buildings and groups of Kac-Moody type. In: Groups, combinatorics & geometry (Durham, 1990), volume 165 of London Math. Soc. Lecture Note Ser., 249–286, Cambridge Univ. Press 1992. MR1200265 Zbl 0851.2202310.1017/CBO9780511629259.023Search in Google Scholar
[13] J. Tits, Résumés des cours au Collège de France 1973–2000, volume 12 of Documents Mathématiques (Paris). Société Mathématique de France, Paris 2013. MR3235648 Zbl 1286.01001Search in Google Scholar
[14] J. Tits, R. M. Weiss, Moufang polygons. Springer 2002. MR1938841 Zbl 1010.2001710.1007/978-3-662-04689-0Search in Google Scholar
[15] R. M. Weiss, The structure of affine buildings, volume 168 of Annals of Mathematics Studies. Princeton Univ. Press 2009. MR2468338 Zbl 1166.51001Search in Google Scholar
[16] K. Wendlandt, Exceptional twin buildings of type ξC2 PhD thesis, Justus-Liebig-Universität Giessen, 2021.Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Topology of tropical moduli spaces of weighted stable curves in higher genus
- Explicit p-harmonic functions on the real Grassmannians
- Cohomogeneity one central Kähler metrics in dimension four
- Tropical Poincaré duality spaces
- Isometries of wall-connected twin buildings
- On automorphisms of semistable G-bundles with decorations
- Abelian branched covers of rational surfaces
- Polyhedral compactifications, I
Articles in the same Issue
- Frontmatter
- Topology of tropical moduli spaces of weighted stable curves in higher genus
- Explicit p-harmonic functions on the real Grassmannians
- Cohomogeneity one central Kähler metrics in dimension four
- Tropical Poincaré duality spaces
- Isometries of wall-connected twin buildings
- On automorphisms of semistable G-bundles with decorations
- Abelian branched covers of rational surfaces
- Polyhedral compactifications, I
