Contemporary Mathematics Volume 250, 1999 Open problems in combinatorial group theory Gilbert Baumslag Alexei G. Myasnikov Vladimir Shpilrain Introduction This is a collection of open problems in combinatorial group theory, which is based on a similar list available on-line at our web site http:/ jzebra.sci.ccny.cuny.edujweb/ In selecting the problems, our choices have been, in part, determined by our own tastes. In view of this, we have welcomed suggestions from other members of the community. We want to emphasize that many people have contributed to the present collection, especially to the Background part. In particular, we are grateful to R.Gilman, V.Remeslennikov, I.Kapovich, W.Dicks, G.Bergman, G.Conner, V.Roman'kov, and D.Wise for useful com- ments and discussions. Still, we admit that the background we currently have is far from complete, and it is our ongoing effort to make it as complete and comprehensive as possible. We invite an interested reader to check on our on-line list for a latest update. One more thing concerning our policy that we would like to point out here, is that we have decided to keep on our list those problems that have been solved after the first draft of the list was put on-line in June 1997, since we believe those problems are an important part of the list anyway, because of their connections to other, yet unsolved, problems. Solved problems are marked by a*, and a reference to the solution is provided in the background. We have arranged the problems under the following headings: OUTSTANDING PROBLEMS, FREE GROUPS, ONE-RELATOR GROUPS, FINITELY PRESENTED GROUPS, HYPERBOLIC AND AUTOMATIC GROUPS, BRAID GROUPS, NILPOTENT GROUPS, METABELIAN GROUPS, SOLVABLE GROUPS. Disclaimer. We want to emphasize that all references we give and attri- butions we make reflect our personal opinion based on the information we have. In particular, if we are aware that a problem was raised by a specific person, we make mention of that here. We welcome any additional infor- mation and/ or corrections on these issues. 1991 Mathematics Subject Classification: Primary 20-02, 20Exx, 20Fxx, 20Jxx, Sec- ondary 57Mxx. © 1999 American Mathematical Society 1
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