Let $(M,g)$ be a Riemannian manifold with non-negative Ricci curvature. The Bishop-Gromov volume comparison says that: if $$\alpha_M=\lim_{r\rightarrow\infty}\frac{VolB^M(p,r)}{\omega_nr^n},$$ then $0\leq \alpha_M\leq 1$. Now suppose $\Gamma$ be a non-trivial subgroup of $\pi_1(M)$. Consider the covering space $\rho:\bar{M}/\Gamma\rightarrow M$, where $\bar{M}$ is the universal cover of $M$. Take $$\alpha_{\bar{M}/\Gamma}=\lim_{r\rightarrow\infty}\frac{VolB^{\bar{M}/\Gamma}(q,r)}{\omega_nr^n},$$ where $\rho(q)=p$ and $VolB^{\bar{M}/\Gamma}(q,r)$ is the ball in $\bar{M}/\Gamma$. $\bar{M}/\Gamma$ has the pull back metric $\rho^*g$.
Is there any relation between $\alpha_M$ and $\alpha_{\bar{M}/\Gamma}$? It is clear that $0\leq \alpha_{\bar{M}/\Gamma}\leq 1$, but I am not able to find a relation between $\alpha_M$ and $\alpha_{\bar{M}/\Gamma}$.
