Lately, I am trying to understand the 1982 Hamilton's paper tilted "three-manifold with positive Ricci curvature". In order to get a uniformly lower bound for the scalar curvature, he used an argument about the relation between the volume of universal cover and the fundamental group as in the image below, that is $X$ has volume one, then the volume of its universal cover $Y$ is just number of elements in the fundamental group of $X$. 
Anyone can give me proof of this? thank you !!