Baumslag conjectured in the 1970s that the automorphism tower of a finitely generated free group (free nilpotent group) must be very short. Dyer and Formanek [9] justified the conjecture concerning finitely generated free groups in the “sharpest sense” by proving that the automorphism group Aut(Fn) of a non-abelian free group Fn of finite rank n is complete. Recall that a group G is said to be ...

We classify pointed Hopf algebras with finite Gelfand-Kirillov dimension, which are domains, whose groups of group-like elements are finitely generated and abelian, and whose infinitesimal braidings are positive.

A celebrated theorem of Marshall Hall implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela and provides a purely topological proof that f...

If a soluble group G contains two finitely generated abelian subgroups A, B such that the number of double cosets AgB is finite, then G is shown to be virtually polycyclic.

A module over a PID is an abelian group that also carries multiplication by a particularly convenient ring of scalars. Indeed, when the scalar ring is the integers, the module is precisely an abelian group. This writeup presents the structure theorem for finitely generated modules over a PID. Although the result is essentially similar to the theorem for finitely generated abelian groups from ea...

A module over a PID is an abelian group that also carries multiplication by a particularly convenient ring of scalars. Indeed, when the scalar ring is the integers, the module is precisely an abelian group. This writeup presents the structure theorem for finitely generated modules over a PID. Although the result is essentially similar to the theorem for finitely generated abelian groups from ea...

A module over a PID is an abelian group that also carries multiplication by a particularly convenient ring of scalars. Indeed, when the scalar ring is the integers, the module is precisely an abelian group. This writeup presents the structure theorem for finitely generated modules over a PID. Although the result is essentially similar to the theorem for finitely generated abelian groups from ea...

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.

A group Γ is called boundedly generated (BG) if it is the settheoretic product of finitely many cyclic subgroups. We show that a BG group has only abelian by finite images in positive characteristic representations. We use this to reprove and generalise Rapinchuk’s theorem by showing that a BG group with the FAb property has only finitely many irreducible representations in any given dimension ...

Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, Out(Fn), free group rank n is either virtually finitely generated abelian or contains a nonabelian group. In this note we consider more general situation Out(G) free-by-finite G. We show translation discrete