2 spewwing

This is sort of a sideways look at your question:

There's software called "Heegaard" by John Berge that takes as input a finite presentation and attempts to find a corresponding Heegaard splitting of a 3-manifold which has that fundamental group. It seems to be fairly effective. There are algorithms to produce triangulations of Heegaard splittings available (Hall and Schleimer for example). So you could take the presentation, find (if possible) the Heegaard splitting, produce the triangulation and them then use software like Regina and SnapPea to analyze the geometry of those manifolds. There's a lot of heuristics there and also some serious algorithms. All the links the the various packages and their documentation is available here: http://www.math.uiuc.edu/~nmd/computop/

So for groups that are the fundamental groups of 3-manifolds at least, there's a decent toolkit to play with.

As an example, consider testing to see if a group is trivial. Step 1: Heegaard could get stuck. Step 2: if Heegaard finds a splitting, you triangulate it and pass it to Regina. Step 3: Regina has an algorithm to recognise a triangulated 3-sphere, so it will tell you whether or not your group is trivial.

1

This is sort of a sideways look at your question:

There's software called "Heegaard" by John Berge that takes as input a finite presentation and attempts to find a corresponding Heegaard splitting of a 3-manifold which has that fundamental group. It seems to be fairly effective. There are algorithms to produce triangulations of Heegaard splittings available (Hall and Schleimer for example). So you could take the presentation, find (if possible) the Heegaard splitting, produce the triangulation and them use software like Regina and SnapPea to analyze the geometry of those manifolds. There's a lot of heuristics there and also some serious algorithms. All the links the the various packages and their documentation is available here: http://www.math.uiuc.edu/~nmd/computop/

So for groups that are the fundamental groups of 3-manifolds at least, there's a decent toolkit to play with.

As an example, consider testing to see if a group is trivial. Step 1: Heegaard could get stuck. Step 2: if Heegaard finds a splitting, you triangulate it and pass it to Regina. Step 3: Regina has an algorithm to recognise a triangulated 3-sphere, so it will tell you whether or not your group is trivial.