Pointwise stabilizer of an apartment of the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$

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.

• could you please correct your notation? e.g. yout 1st line mentions $\mathbb{Q}$ - should it be $\mathbb{Q}_p$ ? Then, $\mathbb{Z}$ is more or less reserved as notation for the usual integers. – Dima Pasechnik Jun 11 at 17:14
• I already corrected it. Thank you! – user113290 Jun 11 at 17:49
• This is an interesting question. Where does it come from? – LSpice Jun 13 at 16:42
• @LSpice: Is a long story. I am interested on computing certain invariants for S-arithmetic groups, and at some point of my computation I end up facing these pointwise stabilizers of apartments... – user113290 Jun 13 at 19:06