MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 10 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.