What is group manifold of a compact Lie Group? I searched on google the meaning of a group manifold of a compact lie group but I didn't get the answer.
A paper on arxiv "Background Independent Quantum Gravity:A Status Report- Abhay Ashtekar" on the page no. 25 talks about this. 
Can anyone help me?
 A: Take the special unitary group $\operatorname{U}(1)$ for example.
It is a group in that it is a set of elements that satisfy the four axioms of a group. Furthermore it satisfies the axioms that are necessary to make it a Lie Group. From here it is a theorem that: sets with a relation satisfying the axioms for a Lie Group have a natural manifold structure associated with them.
Now let's consider this set's natural manifold structure. Unitary matrices in one dimension have $n^2 = 1$ free parameter. The set of all elements of this group can naturally be given the structure of a circle, the manifold $S^1$. So when we talk about the manifold of a group we are referring to the fact that the elements have a natural manifold structure associated with them, as well as a group structure.
Now referring to the paper you cite, there is a separate thing called an action. This is where the group elements, or points in the group manifold, describe transformations on another space, for example the phase space of a particle. We would colloquially say a 'particle evolving on U(1)'.
A: It should be clear that a group manifold of a compact lie group is the compact lie group because lie groups are manifolds .
