Given a presentation $ < X ; R >$ of a group $G$. Suppose we know for some reason that $G$ is the fundamental group of a three-dimensional finite volume manifold.

Then there is a injective group homomorphism $G\rightarrow Isom(\mathbb{H}^3)$. Mostows rigidity theorem then tells us that it is unique up to composition with inner automorphisms of $\mathbb{H}^3$ from the left (and probably automorphisms of $G$ from the right).

As usual to speficy a homomorphism from a group with a specific presentation to another group we have to pick a image for each generator, such that all relators get mapped to the neutral element. My hope is that there might be an algorithm that starts with some first choice of the images of the generators. Note that $Isom(H^3)$ can be viewed as a subgroup of $GL_4(\mathbb{R})$. Then I would like to iteratively minimize the sum of some norm of the images of the relators (hoping that it will converge to a homomorphism). Of course there are also noninjective homomorphisms (like the trivial one) so I cannot hope that the sequence always converges to an injective homomorphism. But maybe it does with high probability for a reasonably random first choice of generators.

Has someone already done this ? If so I am interested in the convergence properties. They might also depend on the given presentation. So one could also hope that any fundamental group of a hyperbolic 3-manifold has a nice presentation.