Adding two generators and one arbitrary relator to a nontrivial torsion-free group, we always obtain an SQ-universal group. In the course of the proof of this theorem, we obtain some other results of independent interest. For instance, adding one generator and one relator in which the exponent sum of the additional generator is one to a free product of two nontrivial torsion-free groups, we als...

Let G be a group that acts freely on a Λ-tree, where Λ is an ordered abelian group, and let x, y, z be elements in G. We show that if xpyq = zr with integers p, q, r ≥ 4, then x, y and z commute. As a result, the one-relator groups with xpyq = zr as relator, are examples of hyperbolic and CAT(−1) groups which do not act freely on any Λ-tree.

We prove that with probability tending to 1, a 1-relator group with at least 3 generators and the relator of length n is residually finite, virtually residually (finite p)-group for all sufficiently large p, and coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions.

It is known that the word problem for one-relator groups and for one-relator monoids of the form Mon〈A ‖ w = 1〉 is decidable. However, the question of decidability of the word problem for general one-relation monoids of the form M = Mon〈A ‖ u = v〉 where u and v are arbitrary (positive) words in A remains open. The present paper is concerned with one-relator inverse monoids with a presentation o...

In 2006 we completed the proof of a five-part conjecture which was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2-generator, 2-relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theore...

James Howie August 3, 2000 Abstra t A short proof, using graphs and groupoids, is given of Brodski 's Theorem that torsion-free one-relator groups are lo ally indi able 1 Introdu tion In 1980, Sergei Brodski announ ed [2℄ the result, previously onje tured by Gilbert Baumslag [1℄, that every torsion-free one-relator group is lo ally indi able, that is, every nontrivial, nitely generated subgroup...

We investigate the modular group as a finitely presented group. It has a large collection of interesting quotients. In 1987 Conder substantially identified the onerelator quotients of the modular group which are defined using representatives of the 300 inequivalent extra relators with length up to 24. We study all such quotients where the extra relator has length up to 36. Up to equivalence, th...