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Problems in combinatorial group theory. (English) Zbl 0619.20013
Combinatorial group theory and topology, Sel. Pap. Conf., Alta/Utah 1984, Ann. Math. Stud. 111, 3-33 (1987).
[For the entire collection see Zbl 0611.00010.]
The following problems are discussed. Problem 1. Do all nonabelian free groups have the same elementary theory? Problem 2. If G has a presentation \(G=gp(X\|\) R) and \(H=gp(X\cup t\|\) \(R\cup w)\) is obtained by adding one new generator and one defining relation, when does the inclusion \(X\to X\cup t\) induce an injection of G into H? Problem 3. Given an equation over a free group, find an algebraic description of the set of all solutions. Problem 4. Let \(w(a_ 1,...,a_ n)\) be a word in the free group F with basis \(a_ 1,...,a_ n\). Is there an algorithm which, given g in F, decides if there exist \(t_ 1,...,t_ n\) in F such that \(w(t_ 1,...,t_ n)=g?\) Problem 5. Determine the structure of Aut F, of its subgroups, especially its finite subgroups, and its quotient groups, as well as the structure of individual automorphisms. Problem 6. If \(G=F/N\), F free, what subgroup of Aut G are images of subgroups of Aut F? Problem 7. Obtain finite presentations for the mapping class groups that are at once usably concise and yet in which both the generators and the relations have fairly obvious geometrical meanings. Problems 8. Study the structure of the automorphism groups of trees and of their subgroups. Problem 9. The various constructions mentioned in the text appear to open the way to a more comprehensive theory of the structure of infinite groups. Problem 10. Develop a general theory of Burnside groups, including a detailed study of certain particular such groups. Problem 11. Is the conjugacy problem solvable for all one-relator groups? Problem 12. There is clearly much to be done in determining the possible growth functions of groups and in relating them to properties of groups. Problem 13. Extend and relate the theories of the deficiency, the rate of growth, and the Euler Poincaré characteristic. In particular, what influence does the deficiency have on the structure of an infinite group? Problem 14. The Poincaré conjecture. Problem 15. Let the trivial group have a balanced presentation \(gp(X\|\) R), where \(| X| =| R| <\infty\). Can this presentation be reduced to a trivial presentation by a succession of transformations of the following kinds: (i) Nielsen transformations of X, (ii) Nielsen transformations of R, (iii) replacing an element of R by a conjugate, and (iv) Tietze transformations introducing (or deleting) a new element x of X together with a new relator r in R defining x? Problem 16 (the Eilenberg problem). If G has cohomological dimension 2, must it also have geometric dimension 2? Problem 17. Is every subcomplex of an aspherical 2-complex aspherical? Problem 18. Extend small cancellation theory, especially in accordance with the connection or analogy with analysis, and unify the special extensions and applications noted in the text. Problem 19. Develop a unified combinatorial theory of a suitably comprehensive class of ”geometric” groups. Problem 20. Obtain intrinsic descriptions of the Fox subgroups F(n,R) and the dimension subgroups D(n,R) in terms of the commutator structure of the free group F and its normal subgroup R.
”I do not need to tell anyone, - the author says, - that the major source and strength of Combinatorial Group Theory has been Topology, wherein we tentatively include Discontinuous Groups, from Poincaré on, with an assist from the more or less abstract or axiomatic side, beginning with Cayley, through the influence of Finite Groups and of problems from Logic. My account is also biased for the most part toward rather recent work, and to pathways to the future more than monuments to the past.”
Reviewer: Yu.I.Merzlyakov

MSC:
20F05 Generators, relations, and presentations of groups
20-02 Research exposition (monographs, survey articles) pertaining to group theory
00A07 Problem books
20F06 Cancellation theory of groups; application of van Kampen diagrams
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F65 Geometric group theory