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, p-irreducibility and f-irreducibility 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 p-irreducible 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 p-irreducible case. 4. The f-irreducible 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

Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups

We study a class of Labelled Oriented Graph (LOG) group where the underlying graph is a tadpole graph. We show that such a group is the natural HNN extension of a cyclically presented group and investigate the relationship between the LOG group and the cyclically presented group. We relate the second homotopy groups of their presentations and show that hyperbolicity of the cyclically presented group implies solvability of the conjugacy problem for the LOG group. In the case where the label on the tail of the LOG spells a positive word in the vertices in the circuit we show that the LOGs and groups coincide with those considered by Szczepa�nski and Vesnin. We obtain new presentations for these cyclically presented groups and show that the groups of Fibonacci type introduced by Johnson and Mawdesley are of this form. These groups generalize the Fibonacci groups and the Sieradski groups and have been studied by various authors. We continue these investigations, using small cancellation and curvature methods to obtain results on hyperbolicity, automaticity, SQ-universality, and solvability of decision problems

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

We study a class M of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature. By analysing their shift extensions we show that the groups in the class M are are coherent, sub-group separable, satisfy the Tits alternative, possess finite index subgroups of geometric dimension at most two, and that their finite subgroups are all meta-cyclic. Many of the groups in M are virtually free, some are free products of metacyclic groups and free groups, and some have geometric dimension two. We classify the finite groups that occur in M, giving extensive details about the metacyclic structures that occur, and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three. We show that any finite group in the class M that has fixed point free shift automorphism must be cyclic

Efficient Finite Groups Arising in the Study of Relative Asphericity

We study a class of two-generator two-relator groups, denoted Jn(m, k), that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature as finite groups of intriguing orders. Here we find infinite families of non-elementary virtually free groups and of finite metabelian non-nilpotent groups, for which we determine the orders. All Mersenne primes arise as factors of the orders of the non-metacyclic groups in the class, as do all primes from other conjecturally infinite families of primes. We classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups Fa,b,c with trivalent Cayley graphs that was introduced by C.M.Campbell, H.S.M.Coxeter, and E.F.Robertson. The theory of cyclically presented groups informs our methods and we extend part of this theory (namely, on connections with polynomial resultants) to ?bicyclically presented groups? that arise naturally in our analysis. As a corollary to our main results we obtain new infinite families of finite metacyclic generalized Fibonacci groups

Independence property and hyperbolic groups

We prove that existentially closed $CSA$-groups have the independence property. This is done by showing that there exist words having the independence property relatively to the class of torsion-free hyperbolic groups.Comment: v3: 10 pages (11pt), a few typos corrected, minor rearrangements (e.g. Fact 2.3 and Lemma 2.5); v2: 8 pages (10pt), a false statement in the proof of Fact 2.4 is replaced with a true one; v1: 8 page