Using Magnus Representations to Study Automorphism Groups of Free Groups for Both the Study of Mapping Class Groups and Galois Theory, it is Useful to have Tools to Study Automorphism Groups of Free Groups. We Will Discuss One Particular Such Tool Called the Magnus Representation. The Magnus Representation Sends an Automorphism of a Free Group to a Jacobian Matrix of Non-abelian Partial Derivatives, Called Fox Derivatives. For Example, the Magnus Representation for a Genus One Curve with a Rational Point Generalizes the Tate Module Representations from the Theory of Elliptic Curves.

Dr. Rachel Davis
University of Wisconsin-Madison

“For both the study of mapping class groups and Galois theory, it is useful to have tools to study automorphism groups of free groups. We will discuss one particular such tool called the Magnus representation. The Magnus representation sends an automorphism of a free group to a Jacobian matrix of non-abelian partial derivatives, called Fox derivatives. For example, the Magnus representation for a genus one curve with a rational point generalizes the Tate module representations from the theory of elliptic curves”.

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