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?



Question on affine buildings
This is exactly Theorem 8.9 from this paper :


*Werner Ballmann and Michael Brin*, [**Orbihedra of nonpositive curvature**](http://www.numdam.org/item?id=PMIHES_1995__82__169_0), *Inst. Hautes \'Etudes Sci. Publ. Math.* (1995), no. 82, 169--209 (1996). 


