The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani– Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.

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...

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.

In the theory of one-relator groups, Magnus subgroups, which are free subgroups obtained by omitting a generator that occurs in the given relator, play an essential structural role. In a previous article, the author proved that if two distinct Magnus subgroups M and N of a one-relator group, with free bases S and T are given, then the intersection of M and N is either the free subgroup P genera...

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.

Given a one-relator group, we give an effective estimate on the minimal index of a subgroup with a nonabelian free quotient. We show that the index is bounded by a polynomial in the length of the relator word. We also provide a lower bound on the index.

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...

‎‎in cite{fr 1‎, ‎fr 2} using faithful complex representations of cyclically pinched and conjugacy pinched one-relator groups we proved that any limit group has a faithful representation in $psl(2,c)$‎. ‎further this representation can be effectively constructed using the jsj decomposition‎. ‎in this note we show that any hyperbolic cyclically pinched one-relator group with maximal amalgamated ...

We prove that the amalgamated product of free groups with cyclic amalgamations satisfying certain conditions are virtually free-by-cyclic. In case the cyclic amalgamated subgroups lie outside the derived group such groups are free-by-cyclic. Similarly a one-relator HNN-extension in which the conjugated elements either coincide or are independent modulo the derived group is shown to be free-by-c...

The main aim of this paper is to show that every two-generator subgroup of any one-relator group with torsion is either a free product of cyclic groups or is a one-relator group with torsion. This result is proved by using techniques for reducing pairs of elements in certain HNN groups. These techniques not only apply to one-relator groups with torsion but also to a large number of other groups...