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? Background

(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?

*(OR16)  Let S_n  be the orientable surface group of genus  n.
(a) Are the groups S_n and S_m (m,n >1) elementarily equivalent?  (i.e., Th(S_m) =Th(S_n)?)

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