I was reading about homogeneous space i.e let $G$ be a Lie group and $\Gamma$ be a discrete subgroup of $G$ with finite covolume. Then Hausdroff dimension of $G / \Gamma$ is the Hausdorff dimension of $G$. I didn't find any proof of the above fact, if someone has any idea about how to prove it or any reference, it would be very helpful. In fact I want to know the proof also for the fact that Hausdroff dimension of $G$= topological dimension of $G$.
Any idea or reference where Hausdorff dimension has been calculated for the groups would be very helpful for my understanding purposes.