We show how a positive solution to the word problem for finitely generated commutative semigroups leads to a positive solution to the conjugacy problem for a class of HNN groups. Here we answer a question of Professor Wilhelm Magnus concerning the solvability of the conjugacy problem of the groups VA discussed in [1]. These groups are given by (I) <tf„ . . . , ak, b; af^b"'ax = bq\ . . . , ak^b...

If a non-trivial subgroup A of the group of continuous automorphisms of a noncyclic free pro-p group F has finite order, not divisible by p, then the group of fixed points FixF (A) has infinite rank. The semi-direct product F>!A is the universal p-Frattini cover of a finite group G, and so is the projective limit of a sequence of finite groups starting with G, each a canonical group extension o...

Asymptotic properties of finitely generated subgroups of free groups, and of finite group presentations, can be considered in several fashions, depending on the way these objects are represented and on the distribution assumed on these representations: here we assume that they are represented by tuples of reduced words (generators of a subgroup) or of cyclically reduced words (relators). Classi...

We prove that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof: the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F ). Introduction In 1975 J. Dyer and E. Formanek [2] had proved that the automorphism group of a finitely generated non-abelian free group F is complete (that i...

We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups. Mathematics Subject Classification (2000): 20E05 Let F be a free group. Let H and K be nontrivial finitely generated subgroups of F . It is a theorem of Howson [1] that H ∩K has finite rank. H. Neumann proved in [2] that rank(H ∩K)− 1 ≤ ...

In this paper we prove the algorithmic solvability of finite systems of equations over the Q-completion of a torsion-free hyperbolic group. It was recently proved in [1] that finite systems of equations over the Q–completion of a finitely generated free group are algorithmically solvable. In this paper we generalize the results of [1] to the case of the Q–completion G of an arbitrary torsion–fr...

An algorithm is constructed that decides if a given finite system of equations over a free Q-group has a solution, and if it does, finds a solution. 0. Introduction Systems of equations over a group have been widely studied (see, for instance, [4],[5],[11]). This is currently one of the main streams of combinatorial group theory. The problem of deciding if a system of equations in a group has a...

We prove several cases of the following theorem: Every free group word which is not a proper power can represent every permutation of an infinite set. The remaining cases will be proved in a forthcoming paper of R. C. Lyndon. Fx denotes a free group freely generated by the set A. The elements of X are called letters, and the elements of Fx are represented by reduced words in those letters. G de...

An anti-torus is a subgroup 〈a, b〉 in the fundamental group of a compact non-positively curved space X, acting in a specific way on the universal covering space X̃ such that a and b do not have any commuting non-trivial powers. We construct and investigate anti-tori in a class of commutative transitive fundamental groups of finite square complexes, in particular for the groups Γp,l originally st...