Denote by $X$ the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$. Let $\Sigma$ be the fundamental apartment of $X$. Let $\Gamma=\mathrm{SL}_n(\mathbb{Z}[\frac{1}{p}])$.
We can prove that the pointwise stabilizer $\Gamma_\Sigma$ of $\Sigma$ in $\Gamma$ is finite. In fact, $\Gamma_\Sigma$ is the set of diagonal matrices with entries in $\mathbb Z_p\cap \mathbb{Z}[\frac{1}{p}]=\mathbb Z$, where $\mathbb{Z}_p$ are the $p$-adic integers. Therefore $\Gamma_\Sigma$ is isomorphic to $(\mathbb{Z}/2)^n$.
My question is. Is the pointwise stabilizer of any apartment of $X$ in $\Gamma$ finite?
Since $\Gamma$ acts chamber transitively on $X$, it is enough to prove that the stabilizer of an apartment, containing the fundamental chamber, is finite. In this case $\Gamma_\Sigma$ is a subgroup of $\mathrm{SL}_n(\mathbb{Z})$ since the stabilizer in $\Gamma$ of the fundamental chamber is contained in $\mathrm{SL}_n(\mathbb{Z})$. Moreover, $\Gamma_\Sigma$ is abelian since it is subconjugated to the abelian subgroup of diagonal matrices of $\mathrm{SL}_n(\mathbb{Q}_p)$. It is the most I can simplify the problem. I hope I am not too far from the answer.
