Use of Fox Derivatives for Solution of Group Theoretic Problem

Abstract

The thesis is an exposition of use of Fox derivatives for solution of topological and group theoretic problems. Our own contribution is in the latter area. Developed by Fox, free differential calculus emerged as a powerful tool for the solution of topological problems. In addition to reviewing application of this tool in various mathematical situations, (i) we have applied it to provide new proofs of a few very important results in combinatorial group theory, (ii) solved the conjugacy problem for torsion-free polycyclic-by-finite groups and torsion-free groups with single defining relations and (iii) determined homology and cohomology of a few classes of groups. In the first chapter the free differential calculus of Fox has been introduced and its application to classification of topological spaces, has been briefly described. In this context, Lens spaces, knots, links and braids have been defined and use of Alexander polynomials in their classification described A few useful tenns in combinatorial group theory have been defined and a number of fundamental results stated. The second chapter describes Birman's work on the relationship of the Jacobians with automorphisms of free groups. This has been used to determine the automorphism group of a free group of rank 2. The third chapter provides a new proof of Freiheitssatz, a fundamental theorem in the theory of single relator groups. This was first proved by Magnus who used it to solve the word problem for such groups. Many proofs (some of them geometric) of his theorem and its generalisations have so far been given by eminent mathematicians, viz. Lyndon, Schupp, Bums, etc. We proved this theorem using Fox derivatives following Majumdar's ideas. ---------