Lecture Notes 1 - Some Topics of the Combinatorial Group Theory

Some parts of this lecture note are constructed from a course of lectures for the first year postgraduate students at Glasgow University. Duration of the course and so in here, the background level assumed of the student consisted of little more than the fundamentals of groups. Combinatorial group theory is concerned with groups which are given by means of presentations. A rich and fruitful method of studying groups given this way is to use geometric and topological techniques. In literature there are plenty number of sources about Combinatorial Group Theory. We may suggest the references at the last part of this book to the readers for other topics that are not mentioned in here. Therefore this large number of sources effect positively the number of subject that can be studied in postgraduate studies. In this lecture note series book, we will present some special topics which have been sensitively to chosen the material in the meaning of a straight direction for such postgraduate studies. All important materials are given with references and thus it is aimed to guide the reader in their studies. Rather than theoretical approaches, computational methods are given with examples. We strictly note that this note is not enough to produce an academic publication, it is just try to give some fundamental ideas to the reader. However, we should not forget that there are many more open problems that can be solved with the concepts given in this note. Free groups and their properties, presentations of groups, combinatorial 1-complexes, combinatorial 2-complexes are the main contents of this book. End of the book, we will present two Appendix chapters: The first one is dealing with the basements of rewriting systems with a geometric view point while the other is about the classification of groups in terms of their abelianizations by using Tietze transformations and a small part of graphs.