Ramos Cuevas, Carlos (2009): On Convex Subcomplexes of Spherical Buildings and Tits’ Center Conjecture. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

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Abstract
In this thesis we study convex subcomplexes of spherical buildings. In particular, we are interested in a question of J. Tits which goes back to the 50’s, the socalled Center Conjecture. It states that a convex subcomplex of a spherical building is a subbuilding or the building automorphisms preserving the subcomplex have a common fixed point in it. A proof of the Center Conjecture for the buildings of classical types (An, Bn and Dn) has been given by B. Muehlherr and J. Tits in [MT06]. The F4case was presented by C. Parker and K. Tent in a talk in Oberwolfach [PT08]. Both approaches use combinatorial methods from incidence geometry. B. Leeb and the author gave in [LR09] differentialgeometric proofs for the cases F4 and E6 from the point of view of the theory of metric spaces with curvature bounded from above. In this work we develop the differentialgeometric approach further. Our main result is the proof of the Center Conjecture for buildings of type E7 and E8, whose geometry is considerably more complicated. In particular, this completes the proof of the Center Conjecture for all thick spherical buildings. We also give a short differentialgeometric proof for the classical types. Finally, we show how the cases F4, E6 and E7 can be deduced from the E8case.
Item Type:  Thesis (Dissertation, LMU Munich) 

Keywords:  spherical buildings, CAT(1) spaces, convex sets 
Subjects:  600 Natural sciences and mathematics > 510 Mathematics 600 Natural sciences and mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date Accepted:  21. December 2009 
1. Referee:  Leeb, Bernhard 
Persistent Identifier (URN):  urn:nbn:de:bvb:19110050 
MD5 Checksum of the PDFfile:  1169970078aaf51219f20d04a91b10ef 
Signature of the printed copy:  0001/UMC 18278 
ID Code:  11005 
Deposited On:  20. Jan 2010 12:38 
Last Modified:  16. Oct 2012 08:34 