most recent 30 from http://mathoverflow.net 2013-05-25T13:44:35Z http://mathoverflow.net/feeds/question/11026 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11026/automatic-groups-recent-progress Michal Kotowski 2010-01-07T09:36:47Z 2010-01-15T20:07:38Z

<p>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?</p>

http://mathoverflow.net/questions/11026/automatic-groups-recent-progress/11915#11915 Agol 2010-01-15T20:07:38Z 2010-01-15T20:07:38Z

<p>I'm not an expert in the area, but here's a few highlights:</p> <p>Bridson <a href="http://www.ams.org/mathscinet-getitem?mr=2016694" rel="nofollow">distinguished automatic and combable groups.</a> </p> <p>Burger and Mozes found examples of <a href="http://www.ams.org/mathscinet-getitem?mr=1839489" rel="nofollow">biautomatic simple groups.</a> </p> <p>Mapping class groups were originally shown to be automatic by Mosher. Recently, Hamenstadt has shown that <a href="http://front.math.ucdavis.edu/0912.0137" rel="nofollow">they are biautomatic</a>.</p> <p>See part 3 of McCammond's survey for an <a href="http://aimath.org/pggt/Decision%5FProblems" rel="nofollow">update on open problems</a>. </p> <p>A well-known open problem is whether automatic groups have solvable conjugacy problem.</p>