Epstein's (et al.) "Word Processing in Groups" is a quite comprehensive monograph on automatic groups, finite automata in geometric group theory, specific examples like braid groups, fundamental groups of 3-dim manifolds etc. However, the book is from 1992, so much of the material summarizes research done by Cannon, Thurston, Holt etc. back in the '80s. I'm interested in how the theory of automatic groups (and, more generally, applications of formal languages in group theory) has progressed since then - have there been any significant new results, open problems, novel ideas, examples?
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I'm not an expert in the area, but here's a few highlights:
Bridson distinguished automatic and combable groups.
Burger and Mozes found examples of biautomatic simple groups.
Mapping class groups were originally shown to be automatic by Mosher. Recently, Hamenstadt has shown that they are biautomatic.
See part 3 of McCammond's survey for an update on open problems.
A well-known open problem is whether automatic groups have solvable conjugacy problem.
answered Jan 15, 2010 at 20:07
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$\begingroup$ The link given here seems to be dead: "McCammond's survey for an update on open problems." And I did not get the pdf file from the Wayback Machine either. Is the paper in question perhaps Algorithms, Dehn Functions and Automatic Groups? $\endgroup$ Commented Jan 13, 2022 at 8:17
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$\begingroup$ @MartinSleziak Hmm, that looks that’s it. It looks like they took the wiki down, and it resides here: sites.google.com/a/scu.edu/rscott/pggt You need permission to access the google drive, so I’m waiting to be granted access. $\endgroup$– Ian AgolCommented Jan 13, 2022 at 17:46