I'm trying to see how lattices can be defined in terms of the volume of the associated locally symmetric space. In Exercise §1C#6 of Dave Witte Morris book, one is asked to prove the following:
Let $K$ be a compact subgroup of a Lie group $G$, and $Γ$ be a discrete subgroup of $G$ that acts freely on $G/K$.
Show that $Γ \backslash G$ has finite volume if and only if $Γ \backslash G/K$ has finite volume.
I don't quote see how to solve this. To begin with, shouldn't $(G,K)$ be a Riemannian symmetric pair in order for $\Gamma \backslash G/K$ to have the structure of locally symmetric space and thus have a Riemannian metric?