Hi! I'm trying to understand why a hyperbolic n-manifold has finite mapping class group if $n \geq 3 $. In books I'm reading it's said it's a consequence of Mostow's rigidity theorem:
"If M and N are complete hyperbolic manifolds with finite total volume, any isomorphism of fundamental groups is realized by a unique isometry."


A corollary of this is that:
" If M is hyperbolic (complete, with finite total volume) and $n \geq 3 $, then Out($\pi\_{1}(M)$) is a finite group, isomorphic to the group of isometries of M ".


But how could this could solve my problem? I mean, I know there is Dehn-Nielsen Theorem which states that Out($\pi\_{1}(M)$) is isomorphic to MCG(M), but I know this to be true only in dimension 2...what can I say in dimension (at least) 3?
Thank you.



Why is the mapping class group of hyperbolic manifolds finite?
In dimension three, this was proven by [Gabai, Meyerhoff, and N. Thurston](http://www.ams.org/mathscinet-getitem?mr=1973051). This was proven orginally for Haken 3-manifolds by Hatcher. Gabai extended this to hyperbolic 3-manifolds satisfying a certain technical condition, which was then verified for all closed hyperbolic 3-manifolds in the above paper. Gabai extended this result to prove that $Diff(M) \simeq Isom(M)$. 


The analogous result in higher dimensions was proved by [Farrell and Jones](http://www.ams.org/journals/bull/1988-19-01/S0273-0979-1988-15640-6/home.html) (see Theorem 5, I think this is only for dimension $>5$, but this isn't explicitly stated). Proofs are given [here](http://www.ams.org/mathscinet-getitem?mr=973309). I don't think dimensions $4$ or $5$ have been worked out. 


