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Completely reducible subcomplexes of spherical buildings. (English) Zbl 1226.51003

In 2005, J.-P. Serre [Astérisque 299, Exp. No. 932, 195–217 (2005; Zbl 1156.20313)] formulated the “Centre Conjecture”, which he attributed to Tits. This conjecture is about a convex subcomplex \(\Omega\) of a spherical building \(\Delta\): either there is a simplex (“centre”) in \(\Omega\) fixed by any automorphism of \(\Delta\) fixing \(\Omega\), or, for every simplex \(A\) in \(\Omega\), there is another simplex in \(\Omega\) which is opposite to \(A\). (This last possibility implies in fact that \(\Omega\) itself is a building). The centre conjecture was proved for classical buildings by B. Mühlherr and J. Tits [J. Algebra 300, No. 2, 687–706 (2006; Zbl 1101.51004)].
In this paper, the authors give another, very short proof of the conjecture for classical (i.e. of type \(A_n\), \(B_n\), \(C_n\) or \(D_n\)) buildings.

MSC:

51E24 Buildings and the geometry of diagrams
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References:

[1] Michael Bate, Benjamin Martin, Gerhard Röhrle: On Tits’ centre conjecture for fixed point subcomplexes. C. R. Math. Acad. Sci. Paris 347, 353–356 (2009) · Zbl 1223.20041 · doi:10.1016/j.crma.2009.02.018
[2] B. Leeb and C. Ramos-Cuevas, The center conjecture for spherical buildings of types F4 and E6, $${\(\backslash\)tt arXiv:0905.0839v2}$$ . · Zbl 1232.51008
[3] Mühlherr B, Tits J.: The center conjecture for non-exceptional buildings. J. Algebra 300, 687–706 (2006) · Zbl 1101.51004 · doi:10.1016/j.jalgebra.2006.01.011
[4] C. Parker and K. Tent, Convexity in buildings, in: Buildings: interactions with algebra and geometry. Abstracts from the workshop held January 20–26, 2008, Oberwolfach Rep. 5 (2008), 119–172.
[5] C. Ramos-Cuevas, The center conjecture for thick spherical buildings, $${\(\backslash\)tt arXiv:0909.2761v1}$$ . · Zbl 1283.51006
[6] L. Kramer, A completely reducible subcomplex of a spherical building is a spherical building, $${\(\backslash\)tt arXiv:1010.0083v1}$$ .
[7] J.-P. Serre, Complète réductibilité, Séminaire Bourbaki. Vol. 2003/2004, Astérisque 299, 2005.
[8] J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, 386, Springer-Verlag, Berlin, 1974. · Zbl 0295.20047
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