We consider the following conjecture. Suppose that G is a non-free non-cyclic onerelator group such that each subgroup of finite index is again a one-relator group and each subgroup of infinite index is a free group. Must G be a surface group? We show that if G is a freely indecomposable fully residually free group and satisfies the property that every subgroup of infinite index is free then G ...

We prove that for " random " one-relator groups the Delzant T-invariant (which measures the smallest size of a finite presentation of a group) is comparable in magnitude with the length of the defining relator. The proof relies on our previous results regarding isomorphism rigidity of generic one-relator groups and on the methods of the theory of Kolmogorov-Chaitin complexity.

If S is noncompact, or has nonempty boundary, then π1(S) is free, and the answer to Question 1 is yes, by an old result of Magnus [7] on one-relator groups. (Essentially, the defining relator in a one-relator group on a given generating set is unique up to conjugacy and inversion.) We will show (see Theorem 3.4 below) that Question 1 also has an affirmative answer in the case of a closed surfac...

A recent result of D. J. Collins states that the intersection of two Magnus subgroups in a one-relator group consists either just of the ‘obvious’ intersection, or exceptionally of the free product of the obvious intersection with a free group of rank 1. In this paper, Collins’ result is generalised to apply to one-relator products of arbitrary locally-indicable groups. Moreover, a precise anal...

the freiheitssatz of magnus for one-relator groups is one of the cornerstones of combinatorial group theory. in this short note which is mostly expository we discuss the relationship between the freiheitssatz and corre-sponding results in free power series rings over fields. these are related to results of schneerson not readily available in english. this relationship uses a faithful representa...

Conjugacy separability of any group of the class of one-relator groups given by the presentation a, b; a m , b n 1m, n > 1 is proven. The proof made used of theoretical combinatorial group methods, namely the structure of amalgamated free products and some properties of the subgroups and quotients of any group of the class of one-relator groups given above.

We introduce conditions on a group action on a tree that are sufficient for the action to extend to the automorphism group. We apply this to two different classes of one-relator groups: certain BaumslagSolitar groups and one-relator groups with centre. In each case we derive results about the automorphism group, and deduce that there are relatively few outer automorphisms.

We prove a Freiheitssatz for one-relator products of torsion-free groups, where the relator has syllable length at most [Formula: see text]. This result applications to equations over groups: in particular singular equation text] group solution some overgroup.

Let G D ha; b; : : : j r D 1i be a one-relator group equipped with at least two generators. For all w which do not commute with r in the ambient free group on the generators a, b, ..., the groups G.r;w/ D ha; b; : : : j rrw D r2i are not residually finite and have the same finite images as G. The existence of this family of one-relator groups which are not residually finite reinforces what is b...

We prove that for " most " one-relator groups Delzant's T-invariant (which measures the smallest size of a finite presentation for a group) is comparable in magnitude with the length of the defining relator. The proof relies on our previous results regarding algebraic rigidity of generic one-relator groups and on the methods of algorithmic probability involving Kolmogorov-Chaitin complexity.