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The center conjecture for convex subcomplexes of spherical buildings

Let Δ\Delta be a (weak) spherical building and let KK be a convex chamber subcomplex of Δ\Delta. A chamber DD is opposite to a chamber CC if they are opposite in the spherical building.

Center conjecture. At least one of the following two possibilities holds:

  1. For each chamber CC in KK there is a chamber DD in KK opposite to CC.
  2. The group of automorphisms of Δ\Delta stabilizing KK stabilizes a non-trivial simplex of KK.

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 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The Center Conjecture for convex subcomplexes of spherical buildings

    Let BB be a spherical building and let K⊆BK\subseteq B be a convex subcomplex. Center Conjecture. Then KK is a subbuilding or the action

    StabAut(B)(K)↷KStab_{Aut(B)}(K)\curvearrowright K

    of the automorphisms of BB preserving KK 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 22, 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).

Sources & referencesView supporting material

Primary source

Koen Struyve, “(Non-)completeness of R-buildings and fixed point theorems”, arXiv:0909.3202 (2009).

fixed-point-theoremsgeometric-group-theoryspherical-buildings

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