See also problems (O4),
(05), (06) in the OUTSTANDING
PROBLEMS section.

(OR1) (G.Baumslag) Are all one-relator groups with torsion residually finite? Background

(OR2) Is the isomorphism problem solvable for one-relator groups with torsion? Background

(OR3) (A.Miasnikov) Is the complexity of the word problem for every one-relator group quadratic, i.e., is there for every one-relator group an algorithm solving the word problem in quadratic time with respect to the length of a word? In polynomial time?

(OR4) Is the subgroup membership problem solvable for one-relator groups? That is, is there an algorithm for deciding if a given element of the group belongs to a given finitely generated subgroup?

(OR5) Is it true that if the relation module of a group G is cyclic, then G is a one-relator group? Background

(OR6) (G.Baumslag) Let H = F/R be a one-relator group, where R is the normal closure of an element r \in F. Then, let G = F/S be another one-relator group, where S is the normal closure of s = r^k for some integer k. Is G residually finite whenever H is ?

(OR7) (G.Baumslag) Let G = F/R be a one-relator group with the relator from [F,F].

(a) Is G hopfian ?

*(b) Is G residually finite ?

(c) Is G automatic ? Background

(OR8) (G.Baumslag) The same as (OR7), but for a relator of the form [u,v]. Background

*(OR9) (D.Moldavanskii) Are two one-relator groups isomorphic if each of them is a homomorphic image of the other? Background

(OR10) Is every one-relator group without metabelian subgroups automatic? Background

(OR11) (C.Y.Tang) Are all one-relator groups with torsion conjugacy separable?

(OR12) Are all freely indecomposable one-relator groups with torsion co-hopfian?

(OR13) (a) Which finitely generated one-relator groups have all generating systems (of minimal cardinality) Nielsen equivalent to each other ?

(b) Which finitely generated one-relator groups have only tame automorphisms (i.e., automorphisms induced by automorphisms of the ambient free group) ?

(OR14) (G.Baumslag) Describe one-relator groups which are discriminated by a free group. Background

(OR15) (B.Fine) If G is a one-relator group with the property that every subgroup of finite index is again a one-relator group, and every subgroup of infinite index is free, must G be a surface group?

(b) Is S_m elementary equivalent to F_{2m}, the free group
of rank 2m ? Background