Let $ M $ be a compact connected manifold with $$ M \cong \Gamma \backslash G /H $$ where $G $ is a Lie group, $ H $ a compact subgroup, $\Gamma $ a discrete subgroup, and $ G/H $ is connected and simply connected. Then $ \Gamma $ must be the fundamental group of $M$.

1. What else can we say about uniqueness of $G, H $ and $ \Gamma$ ?

2. In the 8 model 3 manifold geometries none of the $ G $ are connected. If $ G $ acts transitively on a connected manifold then so does it’s identity component, so why not just take $ G $ to be connected?

3. Are there canonical choices for $ G, H $ with particularly nice properties, at least in some cases? For example, can $ G/H $ be chosen to be a symmetric space?

Let $ M $ be a compact connected manifold with $$ M \cong \Gamma \backslash G /H $$ where $G $ is a Lie group, $ H $ a compact subgroup, $\Gamma $ a discrete subgroup, and $ G/H $ is connected and simply connected. Then $ \Gamma $ must be the fundamental group of $M$.

1. What else can we say about uniqueness of $G, H $ and $ \Gamma$ ?

2. In the 8 model 3-manifold geometries none of the $ G $ are connected. If $ G $ acts transitively on a connected manifold then so does its identity component, so why not just take $ G $ to be connected?

3. Are there canonical choices for $ G, H $ with particularly nice properties, at least in some cases? For example, can $ G/H $ be chosen to be a symmetric space?