On Convex Subcomplexes of Spherical Buildings and Tits’ Center Conjecture

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 so-called 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 F4-case 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
differential-geometric 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 differential-geometric
proof for the classical types. Finally, we show how the cases F4,
E6 and E7 can be deduced from the E8-case.