Let $X$ be an affine building. Assume that $X$ is periodic, by which I mean that there exists a covering $X\to F$ of a finite simplicial complex. Let $\Gamma$ denote the group of deck transformations, then $F=\Gamma\backslash X$. Call an apartment $A$ periodic, if $\Gamma_A\backslash A$ is compact, where $\Gamma_A$ is the group of $\gamma\in\Gamma$ which satisfy $\gamma A=A$. Is it true that for any two chambers $C,D$ there exists a periodic apartment containing both?
2 Answers
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This is exactly Theorem 8.9 from this paper :
Werner Ballmann and Michael Brin, Orbihedra of nonpositive curvature, Inst. Hautes \'Etudes Sci. Publ. Math. (1995), no. 82, 169--209 (1996).
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May be this is more a comment than an answer, but I cannot post comments.
I don't really know the answer, but before asking for a periodic apartment containing prescribed chambers you must know that there are indeed periodic apartments. This reference by Corina Ciobotaru might be interesting in that respect: http://arxiv.org/abs/1402.5554
But you might already know that reference.
Best Pierre
