We show that the " hard " part of Whitehead's algorithm for solving the automorphism problem in a fixed free group F k terminates in linear time (in terms of the length of an input) on an exponentially generic set of input pairs and thus the algorithm has strongly linear-time generic-case complexity. We also prove that the stabilizers of generic elements of F k in Aut(F k) are cyclic groups gen...

Conjugacy separability of any group of the class of one-relator groups given by the presentation 〈a, b; [am, bn] = 1〉 (m, n > 1) is proven.

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...

Magnus’ Freiheitssatz [20] states that if a group is defined by presentation with m generators and single cyclically reduced relator, this relator contains the last generating letter, then first $$m-1$$ letters freely generate free subgroup. We study an analogue of theorem in Gromov density model random groups [14], showing phase transition phenomenon at $$d_r = \min \{\frac{1}{2}, 1-\log _{2m-...

We apply the method of Arzhantseva-Ol’shanskii to prove that for an exponentially generic (in the sense of Ol’shanskii) class of one-relator groups the isomorphism problem is solvable in at most exponential time. This is obtained as a corollary of the more general result that for any fixed integers m > 1, n > 0 there is an exponentially generic class of m-generator n-relator groups where every ...

It is proven that the Freiheitssatz holds for all one-relator products of cyclic groups if the relator is cyclically reduced and a proper power. The method of proof involves representing such groups in PSL2(C) and is a refinement of a technique of Baumslag, Morgan and Shalen. The technique allows the extension of the Freiheitssatz result to many additional one-relator products.

In this paper we prove several results regarding decidability ofthe membership problem for certain submonoids in amalgamated free prod-ucts and HNN extensions of groups. These general are then appliedto solve the prefix a number classes one-relatorgroups which low Magnus–Moldavanski i hierarchy. Since pre-fix one-relator groups is intimately related to theword special inverse monoids theE-unita...

We show that a class of cyclically pinched one-relator groups admits amenable, faithful and transitive actions on infinite countable sets. This work generalizes the results on such actions for doubles of free group on 2 generators.