Open problems in combinatorial group theory. Second edition Gilbert Baumslag Alexei G. Myasnikov Vladimir Shpilrain Contents 1 Outstanding Problems 2 2 Free Groups 7 3 One-relator Groups 15 4 Finitely Presented Groups 17 5 Hyperbolic and Automatic Groups 20 6 Braid Groups 22 7 Nilpotent Groups 24 8 Metabelian Groups 25 9 Solvable Groups 26 10 Groups of Matrices 27 11 Growth 28 Introduction This is a collection of open problems in combinatorial group theory, which is based on a similar list available online at our web site http://www.grouptheory.org 2000 Mathematics Subject Classi¯cation: Primary 20-02, 20Exx, 20Fxx, 20Jxx, Sec- ondary 57Mxx. 1 In three years since the ¯rst edition of this collection was published in [18], over 40 new problems have been added, and, more importantly, 20 problems have been solved. We hope that this is an indication of our work being useful. In the present edition we have added two new sections: on Groups of Matrices and on Growth. 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 G. Bergman, G. Conner, W. Dicks, R. Gilman, I. Kapovich, V. Remeslennikov, V. Roman'kov, E. Ventura and D.Wise for useful comments and discussions. Still, we admit that the background we currently have is far from complete, and it is our ongoing e®ort to make it as complete and comprehensive as possible. We invite an interested reader to check on our online 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 ¯rst draft of the list was put online 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, Groups of Matrices, Growth. Disclaimer. We want to emphasize that all references we give and attribut- ions 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 speci¯c per- son, we make mention of that here. We welcome any additional information and/or corrections on these issues. 1 Outstanding Problems (O1) The Andrews-Curtis conjecture. Let F = Fn be the free group of a ¯nite rank n ¸ 2 with a ¯xed set X = fx1; :::; xng of free generators. A set Y = fy1; :::; yng of elements of F generates the group F as a normal subgroup if and only if Y is Andrews-Curtis equivalent to X, which means one can get from X to Y by a sequence of Nielsen transformations together with conjugations by elements of F . This problem is of interest in topology as well as in group theory. A topo- logical interpretation of this conjecture was given in the original paper by 2 J. Andrews and M. Curtis [3]. A more interesting topological interpretation arises when one allows one more transformation, \stabilization", when Y is extended to fy1; :::; ym; xºg, where xº is a new free generator (i.e., y1; :::; ym do not depend on xº), and the converse of this transformation. Then the Andrews-Curtis conjecture is equivalent to the following (see [184]): two contractible 2-dimensional polyhedra P and Q can both be embedded in a 3-dimensional polyhedron S so that S geometrically contracts to P and Q. Note that this is true if 3 is replaced by 4 { this follows from a result of Whitehead. The problem is amazingly resistant; very few partial results are known. A good group-theoretical survey is [32]. For a topological survey, we refer to [80]. The prevalent opinion is that the conjecture is false; however, not many potential counterexamples are known. Two of them are given in the survey [32] by R. Burns and O. Macedonska; a one-parameter family of potential counterexamples appears in [1]. Recently, a rather general series of potential counterexamples in rank 2 was reported in [142]. Further potential coun- terexamples are given in [146]. It might be of interest that, by using genetic algorithms, A. D. Myas- nikov and A. G. Myasnikov [145] showed that all presentations of the trivial group with the total length of relators up to 12 satisfy the Andrews-Curtis conjecture. Finally, we mention a positive solution of a similar problem for free solv- able groups by A. G. Myasnikov [144]. (O2) The Burnside problem. For what values of n are all groups of exponent n locally ¯nite? Of particular interest are n = 5; n = 8; n = 9 and n = 12 { values for which, by the experts' opinion, groups of exponent n have a remote chance of being locally ¯nite. In contrast with the previous problem (O1), the bibliography on the Burnside problem consists of several hundred papers. We only mention here that E. Golod [64] constructed the ¯rst example of a periodic group which is not locally ¯nite; his group however does not have bounded exponent. The ¯rst example of an in¯nite ¯nitely generated group of bounded exponent is due to P. S. Novikov and S. I. Adian [154]. We refer to the book [156] for a survey on results up to 1988, and to the papers [87], [126] for treatment of the most di±cult case where the exponent is a power of 2. We also note that recent work of J. McCammond [133] o®ers a new ap- proach to the Burnside problem via (generalized) small cancellation theory. (O3) Whitehead's asphericity problem. Is every subcomplex of an as- pherical 2-complex aspherical? Or, equivalently: if G = F=R =< x1; :::; xnj r1; :::; rm; ::: > is an aspherical presentation of a group G (i.e., the corre- sponding relation module R=[R; R] is a free ZG-module), is every presenta- tion of the form hx1; :::; xnj ri1 ; :::; rik ; :::i, aspherical as well ? 3 This problem has received considerable attention in the '80s. We mention here a paper by J. Huebschmann [84] that contains a wealth of examples of 2- complexes for which Whitehead's asphericity problem has a positive solution. J. Howie [81] points out a connection between Whitehead's problem and some other problems in low-dimensional topology (e.g. the Andrews-Curtis conjecture). We refer to [123] for more bibliography on this problem. Among more recent papers, we mention a paper by E. Luft [121], where he gives a rather elementary self-contained proof of the main result of Howie's paper (see above) and strengthens the result at the same time. Recently, S. V. Ivanov [90] discovered a connection between Whitehead's asphericity problem and the problem (O12)(a) below. A somewhat more general result was later obtained by I. Leary [112]. (O4) The isomorphism problem for one-relator groups. Z. Sela [168] has solved the isomorphism problem for torsion-free hy- perbolic groups that do not split (as an amalgamated product or an HNN extension) over the trivial or the in¯nite cyclic group. It is not known how- ever which one-relator groups are hyperbolic (cf. problem (O6)). Earlier partial results are [160] and [161]. We also note that two one-relator groups with relators r1 and r2 being isomorphic does not imply that r1 and r2 are conjugate by an automorphism of a free group, which deprives one from the most straightforward way of attacking this problem; see [138]. (O5) The conjugacy problem for one-relator groups. The conjugacy problem for one-relator groups with torsion was solved by B. B. Newman [152]. Some other partial results are known; see e.g. [124] for a survey. (O6) Is every one-relator group without Baumslag-Solitar subgroups hyper- bolic? Note that every one-relator group with torsion is hyperbolic since the word problem for such a group can be solved by Dehn's algorithm { see the paper by B. B. Newman cited in the background to (O5). Therefore, it su±ces to consider torsion-free one-relator groups. We also note that the following weak form of this problem was answered in the a±rmative. A group G is called a CSA group if every maximal abelian subgroup M of G is malnormal, i.e., for any element g in G, but not in M, one has M g \M = f1g. It is known that every torsion-free hyperbolic group is CSA. Now the following weak form of (O6) holds: A torsion-free one-relator group is CSA if and only if it does not contain metabelian Baumslag-Solitar groups BS(1; p) and subgroups isomorp¯c to F2 £ Z [61]. There are also several partial (positive) results on this problem in [91]. 4 (O7) Let G be the direct product of two copies of the free group Fn; n ¸ 2, generated by fx1; :::; xng and fy1; :::; yng, respectively. Is it true that every generating system of cardinality 2n of the group G is Nielsen equivalent to fx1; :::; xn; y1; :::; yng? The importance of this problem is in its relation to two outstanding problems in low-dimensional topology, to the Poincar¶eand Andrews-Curtis conjectures { see e.g.