unit tangent bundle as a quotient of $\pi_{1}(M) \backslash PSL_{2}\mathbb{C}$ with $M$ a compact 3-dimensional hyperbolic manifold

Question

I'm studying the proof of volume rigidity from the Dunfield's paper:"https://arxiv.org/abs/math/9802022". I'm failing to understand the quotient operation in this statement:

Let $G= PSL_{2}\mathbb{C}$, and let $\Gamma = \pi_{1}(M) \subset G$ acting with a discrete faithful representation for M. The unit tangent bundle to M is $X= \Gamma \backslash G$.

Is there someone able to explain me what is going on?

thanks.

Answer 1

The claim is utterly false. Maybe he meant to say $G=PSL(2, {\mathbb R})$, then it is correct (here $M$ is a complete orientable hyperbolic surface). To see that the claim is false as stated, note that $G= PSL(2, {\mathbb C})$ is real 6-dimensional, while the unit tangent bundle of a hyperbolic 3-manifold is 5-dimensional. One can try to remove the adjective "unit" in front of the tangent bundle. Then, at least, dimensions match. But then the manifolds are not even homotopy-equivalent since $TM$ is aspherical while $G$ is homotopy-equivalent to $RP^3$, the real-projective 3-space. However, if he meant to say "oriented orthonormal frame bundle" then his claim is correct.