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I'm new to the combinatorial group theory, so maybe my question is a bit naiive.

I know that the word problem is generally "unsolvable". On the other hand there are specific cases, when the problem can be solved.

It seems that some computer algebra tool, that implements all the progress in that direction would be of use. Do you know any such tool? Or how to use existing CAS'es for this purpose?

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It's not clear what you mean by "all the progress". For example, there is an algorithm to solve the word problem for residually finite groups, but it is so slow that it is worthless in "real life". That being said, most of the useful algorithms (for example, for automatic groups) are implemented in GAP. – Andy Putman Dec 6 '10 at 15:52
I sometimes find Magnus more informative than GAP when dealing with group presentations. But that may be because I don't understand all the relevant GAP features. Magnus is a little more focused on these issues. But I've been feeding the fundamental groups of random 4-manifold triangulations to Magnus and it's been doing pretty well. – Ryan Budney Dec 6 '10 at 15:56
FYI, the easiest-to-install version of Magnus is on the Ubuntu Linux repository. – Ryan Budney Dec 6 '10 at 15:57
A random 4-manifold probably has a hyperbolic fundamental group (it depends on your model for random 4-manifolds, but most models for random group presentation make hyperbolic groups generic), so it shouldn't be too hard to solve the word problem in it. – Andy Putman Dec 6 '10 at 16:08
up vote 6 down vote accepted

Derek Holt's software kbmag (available here) is wonderfully good at solving the word problem of groups given by a finite presentation. Of course, the problem is undecidable in general!

Notice that kbmag can be installed as a GAP package, and then used through sage.

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Sage math seems to use GAP according to the documentation here for word_problem:

Presumably if there were better alternative free word problem solvers then these would have been used instead, because Sage is like the free software borg-of-math.

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