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

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).

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