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I only ask because I don't know how to look for the answer.

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Here's a link to some recent work: people.maths.ox.ac.uk/bridson/papers/BReeves . –  HJRW Dec 17 '12 at 12:03
    
By the way, "I don't know how to look for the answer" suggests a serious obstruction to studying current mathematics. It's not helpful to re-invent the wheel too many times, so you need to develop skills in searching old and new liteature (arXiv included) along with identifying specific people who can provide shortcuts to answers. Traditionally graduate work is supposed to organize this whole complicated process. –  Jim Humphreys Dec 17 '12 at 14:36
    
You do seem to know how to look for an answer: ask on mathoverflow! –  André Henriques Dec 17 '12 at 22:29
    
@Jim: what I mean is, this is the sort of picky question that I'm sure, if settled, was only done recently and I furthermore don't know who would write about it, nor what keywords to put into mathscinet/arxiv/google scholar search etc. Eight years ago or so my first move would have been to ask Dani Wise, just because he taught me about Gromov hyperbolic groups and the Rips complex, but he's not handy right now and I can't get to a library. –  some guy on the street Dec 19 '12 at 3:05
    
@André, you know, I mean it feels a bit like cheating... –  some guy on the street Dec 19 '12 at 3:06

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up vote 7 down vote accepted

It's still an open problem. The isomorphism problem for hyperbolic groups (a much smaller class) was only solved recently by Dahmani and Guirardel (see here), following work of Sela.

In the same vein, the conjugacy problem is also still open for automatic groups. It has been solved for biautomatic groups, but it is also still open whether all automatic groups are biautomatic.

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I would conjecture that the isomorphism problem for automatic groups is undecidable. The class is too large and contains some subclasses where isomorphism problem proved to be very hard, and solutions are quite different. So I would be very surprised if there was a unified solution for all automatic groups. –  Mark Sapir Dec 18 '12 at 10:46

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