the freiheitssatz of magnus for one-relator groups is one of the cornerstones of combinatorial group theory. in this short note which is mostly expository we discuss the relationship between the freiheitssatz and corre-sponding results in free power series rings over fields. these are related to results of schneerson not readily available in english. this relationship uses a faithful representa...

We define and study the Magnus representation for homology cylinders generalizing the work of Kirk, Livingston and Wang [KLW] which treats the case of string links. Using this, we give a factorization formula of Alexander polynomials for three dimensional manifolds obtained by closing homology cylinders. We also show a relationship between the Gassner representation for string links and the Mag...

We consider the Magnus representation of the image of the braid group under the generalizations of the standard Artin representation discovered by M. Wada. We show that the images of the generators of the braid group under the Magnus representation are unitary relative to a Hermitian matrix. As a special case, we get that the Burau representation is unitary, which was known and proved by C. C. ...

We study the Magnus representation for homology cylinders as a generalization of the Gassner representation for string links defined by Le Dimet [12] and Kirk-Livingston-Wang [11]. As an application, we give some factorization formulas of higher-order degree invariants defined by Harvey in [9], [10] for closed three dimensional manifolds obtained from homology cylinders.

From our previous paper, it is known that the Magnus representation of the Torelli group is not faithful. In this paper, we characterize the kernel of its representation for a certain kind of elements.

In this paper, we show that the abelianization of the kernel of the Magnus representation of the automorphism group of a free group is not finitely generated.

ACKNOWLEDGEMENTS I want to thank my adviser, Dr. Magnus Egerstedt, who spurred my interest in graph grammars and modular robotics through his course on networked control. His feedback and guidance throughout my thesis research were invaluable. I also want to thank my family for their continuous support and encouragement in my pursuit of my Master's degree.