OUTSTANDING PROBLEMS

(O1) The Andrews-Curtis conjecture. Let F = F_n be the free group of a finite rank n with a fixed set X = {x_1,...,x_n} of free generators.

(a) A set Y = {y_1,...,y_n} 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.

(b) The same conjecture if one allows one more transformation, "stabilization", where X is extended to {x_1,...,x_n, x_{n+1},...}, and its converse. Background

(O2) The Burnside problem. For what values of n are all groups of exponent n locally finite? 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 finite. Background

(O3) Whitehead's asphericity problem. Is every subcomplex of an aspherical 2-complex aspherical? Or, equivalently: if G = F/R = <x_1,...,x_n | r_1,...,r_m,...> is an aspherical presentation of a group G (i.e., the corresponding relation module R/[R,R] is a free ZG-module), is then every presentation of the form <x_1,...,x_n | r_{i_1},...,r_{i_k},...>, aspherical as well ? Background

(O4) The isomorphism problem for one-relator groups. Background

(O5) The conjugacy problem for one-relator groups. Background

(O6) Is every one-relator group without Baumslag-Solitar subgroups hyperbolic? Background

(O7) Let G be the direct product of two copies of the free group F_n, n \ge 2, generated by {x_1,...,x_n} and {y_1,...,y_n}, respectively.

(a) Is it true that every generating system of cardinality 2n of the group G is Nielsen equivalent to {(x_1,1),...,(x_n,1), (1,y_1),...,(1,y_n)} ?

(b) "Stabilized" version of the same question: Is it true that for every generating system of cardinality 2n, there is k \ge 0 such that this generating system, augmented by (x_{n+1},1),...,(x_{n+k},1), (1,y_{n+1}),...,(1,y_{n+k}) is Nielsen equivalent to {(x_1,1),...,(x_{n+k},1), (1,y_1),...,(1,y_{n+k})} ? Background

*(O8) Tarski's problems. Let F = F_n be the free group of rank n, Th(F) the elementary theory of F, i.e., all sentenses of the group theory language which are true in F.

(a) Is it true that Th(F_2) = Th(F_3) ?

(b) Is Th(F) decidable? Background

*(O9) The Hanna Neumann conjecture. If H and K are nontrivial subgroups of a free group, then rank(H \cap K) - 1 \le (rank(H) - 1)(rank(K) - 1). Background

*(O10) Is the automorphism group of a free group of rank 2 linear? Or, equivalently, is the braid group B_4 linear? Background

(O11) Is there an infinite finitely presented periodic group?

(O12)   (a)  (I.Kaplansky) Can the (integral) group ring of a torsion-free group have zero divisors? Background

(b)  Is it true that the group of units of a group algebra  kG,  where  G  is torsion-free  and  k  is a field,  is generated by  k*  and  G ?