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 ≤ ...

COMMENSURATORS OF RIGHT-ANGLED ARTIN GROUPS AND MAPPING CLASS GROUPS MATT CLAY, CHRISTOPHER J. LEININGER, AND DAN MARGALIT Abstract. We prove that, aside from the obvious exceptions, the mapping class We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to...

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...

We characterise groups in which every abelian subgroup has finite index in its characteristic closure. In a group with this property every subgroup H has finite index in its characteristic closure and there is an upper bound for this index over all subgroups H of G. For every prime p we construct groups G with this property that are infinite nilpotent p-groups of class 2 and exponent p in which...

The paper is devoted to the study of some important types of minimal artinian linear groups. The authors prove that in such classes of groups as hypercentral groups (so also, nilpotent and abelian groups) and FC-groups, minimal artinian linear groups have precisely the same structure as the corresponding irreducible linear groups. 2000 Mathematics Subject Classification. 20E36, 20F28. Let F be ...