You might be interested in this paper of Jason Manning. He proves (assuming the solution to the word problem in $G$---basic undecidability results tell you that it's necessary to assume something like this) that there is an algorithm to construct the discrete faithful hyperbolic representation of $G$.

I don't think the strategy is at all what you suggest, though. Instead, he builds the character variety

$\chi(G)=\mathrm{Hom}(G,SL_2(\mathbb{C}))//SL_2(\mathbb{C})$

and computes its decomposition into irreducibles. Mostow Rigidity implies that a discrete faithful representation is contained in a 0-dimensional component. This gives a finite list of representations to check. Now, for each of these, his algorithm attempts to either construct a fundamental domain using hyperbolic geometry, or to find a proof that the representation is not faithful or discrete.

So one could also hope that any fundamental group of a hyperbolic 3-manifold has a nice presentation.

This sounds extremely optimistic to me. (Well, it depends what you mean by 'nice'.)

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ADDED MUCH LATER

I just learned that another algorithm for finding hyperbolic structures is given in this paper of Luo, Tillmann and Yang.

You might be interested in this paper of Jason Manning. He proves (assuming the solution to the word problem in $G$---basic undecidability results tell you that it's necessary to assume something like this) that there is an algorithm to construct the discrete faithful hyperbolic representation of $G$.

I don't think the strategy is at all what you suggest, though. Instead, he builds the character variety

$\chi(G)=\mathrm{Hom}(G,SL_2(\mathbb{C}))//SL_2(\mathbb{C})$

and computes its decomposition into irreducibles. Mostow Rigidity implies that a discrete faithful representation is contained in a 0-dimensional component. This gives a finite list of representations to check. Now, for each of these, his algorithm attempts to either construct a fundamental domain using hyperbolic geometry, or to find a proof that the representation is not faithful or discrete.

So one could also hope that any fundamental group of a hyperbolic 3-manifold has a nice presentation.

This sounds extremely optimistic to me. (Well, it depends what you mean by 'nice'.)

========

ADDED MUCH LATER

I just learned that another algorithm for finding hyperbolic structures is given in this paper of Luo, Tillmann and Yang.

You might be interested in this paper of Jason Manning. He proves (assuming the solution to the word problem in $G$---basic undecidability results tell you that it's necessary to assume something like this) that there is an algorithm to construct the discrete faithful hyperbolic representation of $G$.

I don't think the strategy is at all what you suggest, though. Instead, he builds the character variety

$\chi(G)=\mathrm{Hom}(G,SL_2(\mathbb{C}))//SL_2(\mathbb{C})$

and computes its decomposition into irreducibles. Mostow Rigidity implies that a discrete faithful representation is contained in a 0-dimensional component. This gives a finite list of representations to check. Now, for each of these, his algorithm attempts to either construct a fundamental domain using hyperbolic geometry, or to find a proof that the representation is not faithful or discrete.

So one could also hope that any fundamental group of a hyperbolic 3-manifold has a nice presentation.

This sounds extremely optimistic to me. (Well, it depends what you mean by 'nice'.)

You might be interested in this paper of Jason Manning. He proves (assuming the solution to the word problem in $G$---basic undecidability results tell you that it's necessary to assume something like this) that there is an algorithm to construct the discrete faithful hyperbolic representation of $G$.

I don't think the strategy is at all what you suggest, though. Instead, he builds the character variety

$\chi(G)=\mathrm{Hom}(G,SL_2(\mathbb{C}))//SL_2(\mathbb{C})$

and computes its decomposition into irreducibles. Mostow Rigidity implies that a discrete faithful representation is contained in a 0-dimensional component. This gives a finite list of representations to check. Now, for each of these, his algorithm attempts to either construct a fundamental domain using hyperbolic geometry, or to find a proof that the representation is not faithful or discrete.

So one could also hope that any fundamental group of a hyperbolic 3-manifold has a nice presentation.

This sounds extremely optimistic to me. (Well, it depends what you mean by 'nice'.)

========

ADDED MUCH LATER

I just learned that another algorithm for finding hyperbolic structures is given in this paper of Luo, Tillmann and Yang.