The center conjecture for convex subcomplexes of spherical buildings
Let be a (weak) spherical building and let be a convex chamber subcomplex of . A chamber is opposite to a chamber if they are opposite in the spherical building.
Center conjecture. At least one of the following two possibilities holds:
- For each chamber in there is a chamber in opposite to .
- The group of automorphisms of stabilizing stabilizes a non-trivial simplex of .
This conjecture is a fixed-point statement for automorphism groups stabilizing convex subcomplexes of spherical buildings. The supplied text identifies it as the center conjecture but gives no evidence of its resolution.
Equivalent formulations 1Other wordings
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
The Center Conjecture for convex subcomplexes of spherical buildings
Let be a spherical building and let be a convex subcomplex. Center Conjecture. Then is a subbuilding or the action
of the automorphisms of preserving has a fixed point. This is a geometric version of Tits' Center Conjecture. The claim is posed here for convex subcomplexes of spherical buildings; the corresponding fixed-point question is known positively in dimensions at most , while the higher-dimensional cases under consideration remain open.
source: B. Leeb and C. Ramos-Cuevas, “The Center Conjecture for spherical buildings of types F4 and E6”, arXiv:0905.0839 (2011).
References
Primary source
Koen Struyve, “(Non-)completeness of R-buildings and fixed point theorems”, arXiv:0909.3202 (2009).
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