If $N$ and $N'$ are two closed hyperbolic 3-manifolds, then one would like to have an algorithm which determines whether or not $N$ and $N'$ are homeomorphic.

If $N$ and $N'$ are Haken, then such an algorithm is provided by Haken. The homeomorphism problem is solved in general by Sela's work on hyperbolic groups and Mostow rigidity.

I was wondering though whether in light of Agol's work one can now give a "purely $3$-dimensional" solution to the homeomorphism problem. More precisely, Agol showed that given any hyperbolic 3-manifold there exists a finite cover with $b_1>0$. So given $N$ and $N'$ I can find a finite cover $\tilde{N}$ with $b_1(\tilde{N})>0$. I can take the subgroup $\pi_1(\tilde{N})$ to be of the type "intersection of all kernels of homomorphisms to a fixed finite group". I can then use the `same' finite index subgroup to get a cover $\tilde{N}'$ with $b_1(\tilde{N}')>0$. So now one can apply Haken to check whether $\tilde{N}$ and $\tilde{N}'$ are homeomorphic. To push this result down to $N$ and $N'$ it seems to me that one needs to calculate the mapping class group of $\tilde{N}$. This group is finite by Mostow rigidity, but are there algorithms for determining it? Are there any bounds on the size of the group?