Title:

On a certain class of cyclically presented groups

Abstract. The present thesis is devoted to the study of a class of cyclically presented groups, an important theme in combinatorial group theory. 1. Introduction We introduce the cyclic presentations which will be the object of our study. We explain why these are important and we state some known results. In view of this we propose a conjecture (Conjecture 1.2.7) to which we give a partial answer in the following chapters. We finally give a definition of irreducibility, pirreducibility and firreducibility for a presentation in the class which we are studying. 2. Method of proof Here we give a short report on split extensions and (van Kampen) diagrams, which are the two basic ingredients involved in our proofs. We then outline the method of proof, which is a generalization of the method used in [11] and makes use of an analysis of modified diagrams. 3. The pirreducible case We start giving a geometric constraint on diagrams and we show, as described in Chapter 2, that a presentation whose diagram respects this constraint gives rise to an infinite group. After studying four particular cases we give conditions on the integer parameters of a presentation in the class considered in order to have a diagram which satisfies the given geometric constraint. Finally, we prove Theorem 2 which partially answers Conjecture 1.2.7 in the pirreducible case. 4. The firreducible case Given certain constraints the problem is reduced to a particular case. We then study this case as outlined in Chapter 2. We also show that these constraints can be weakened if the number of generators is odd. 5. Conclusions We show how the results achieved can be used to prove a theorem in a more general setting; we explain what one should prove in order to confirm Conjecture 1.2.7 and why our method fails in these cases.
