Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G\_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes\_{\mathbb{Q}} F)$. Let $K\_{\infty}$ be a maximal compact subgroup of $G\_{\infty}$. We fix an embedding $G\hookrightarrow GL\_N$ for some $N$. Let $\mathfrak{P}$ denote a prime of $\mathcal{O}\_F$ lying over $p$. Let $G(\mathfrak{P})$ be the intersection of $G\_{\infty}$ with the congruence subgroup of $GL\_N(\mathcal{O}\_F)$ at level $\mathfrak{P}$. Let $\Gamma$ be an arithmetic lattice of $G\_{\infty}$.  

 Let us denote  

 $\Gamma (\mathfrak{P}):=\Gamma \cap G(\mathfrak{P})$.
 Let us denote by $e$ and $f$ the ramification and inertia degrees of $\mathfrak{P}$ in $F$. Hence we get that $[F\_{\mathfrak{P}}:\mathbb{Q}\_p]=ef$ where $F\_{\mathfrak{P}}$ is the completion of $F$ at $\mathfrak{P}$. We define $G\_k:=G \cap (1+p^kM\_{efN}(\mathbb{Z}\_p))$ where $G:=\varprojlim\_k\Gamma /\Gamma({\mathfrak{P}}^k)$ 
 For $k\geqslant 0$, we define $Y\_k:=\Gamma (\mathfrak{P}^{ek})\backslash G\_{\infty} /K\_{\infty}$. 


Then let $vol(\mathfrak{P}^{ek})$ be the volume of $\Gamma(\mathfrak{P}^{ek})\backslash G\_{\infty}$. 


Let $\Gamma(\mathfrak{P}^{ek})\backslash G\_{\infty}$ be compact. 


Then I want to show that $vol(\mathfrak{P}^{ek})\sim [G:G\_k]$. Here $\sim$ denotes that both of them are similar order as $k \rightarrow \infty$. It will be helpful if I can get some hints or some reference on this fact. Thank you in advance.



Volume of arithmetic quotients of symmetric spaces
For a unimodular topological group $G$, and for discrete subgroups $\Theta\subset\Gamma\subset G$, for $f\in C^o\_c(\Theta\backslash G)$, it is true that
$$
\int\_{\Theta\backslash G} f \;=\; \int\_{\Gamma\backslash G}\sum\_{\gamma \in \Theta\backslash\Gamma} f\circ \gamma
$$
in the sense that choice of right $G$-invariant measure on one quotient uniquely determines the right $G$-invariant measure on the other. This follows from the surjectivity of the obvious averaging map from compactly-supported continuous functions on the one quotient to the other.


