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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.

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  • $\begingroup$ 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. $\endgroup$ – Dima Pasechnik Jun 11 at 17:14
  • $\begingroup$ I already corrected it. Thank you! $\endgroup$ – user113290 Jun 11 at 17:49
  • $\begingroup$ This is an interesting question. Where does it come from? $\endgroup$ – LSpice Jun 13 at 16:42
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    $\begingroup$ @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... $\endgroup$ – user113290 Jun 13 at 19:06

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