Largeness and SQ-universality of cyclically presented groups

Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contains, amongst others, the various generalizations of Fibonacci groups introduced by Campbell and Robertson

On the Tits alternative for cyclically presented groups with length four positive relators

We investigate the Tits alternative for cyclically presented groups with length four positive relators in terms of a system of congruences (A),(B),(C) in the defining parameters, introduced by Bogley and Parker. Except for the case when (B) holds and neither (A) nor (C) hold, we show that the Tits alternative is satisfied; in the remaining case we show that the Tits alternative is satisfied when the number of generators of the cyclic presentation is at most 20

On Cyclically Presented Groups of Positive Word Length Four Relators

Substantial progress has been made into the Tits alternative for cyclically presented groups of positive word length four relators. A classif i cation theorem of the i niteness of the abelianisation of said groups is established. A few theorems are proven on the structure of cyclically presented groups of positive word length four for low numbers of generators

Isomorphism theorems for classes of cyclically presented groups

We consider two multi-parameter classes of cyclically presented groups, introduced by Cavicchioli, Repov s, and Spaggiari, that contain many previously considered families of cyclically presented groups of interest both for their algebraic and for their topological properties. Building on results of Bardakov and Vesnin, O'Brien and the previously named authors, we prove theorems that establish isomorphisms of groups within these families

Redundant relators in cyclic presentations of groups

A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain concise refinements of these presentations. We show that the Tits alternative holds for the class of groups defined by redundant cyclic presentations and show that if the number of generators of the cyclic presentation is greater than two, then the corresponding group is large. Generalising and extending earlier results of the authors, we describe the star graphs of orientable and non-orientable cyclic presentations and classify the cyclic presentations whose star graph components are pairwise isomorphic incidence graphs of generalised polygons, thus classifying the so-called (m,k,ν) -special cyclic presentations

Nonhyperbolic free-by-cyclic and one-relator groups

We show that the free-by-cyclic groups of the form F2 ⋊ ℤ act properly cocompactly on CAT(0) square complexes. We also show using generalized Baumslag-Solitar groups that all known groups defined by a 2-generator 1-relator presentation are either SQ-universal or are cyclic or isomorphic to a soluble Baumslag-Solitar group

Some applications of geometric techniques in combinatorial group theory

Abstract available: p. iii-ix

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

Generalized polygons and star graphs of cyclic presentations of groups

Groups defined by presentations for which the components of the corresponding star graph are the incidence graphs of generalized polygons are of interest as they are small cancellation groups that – via results of Edjvet and Vdovina – are fundamental groups of polyhedra with the generalized polygons as links and so act on Euclidean or hyperbolic buildings; in the hyperbolic case the groups are SQ-universal. A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We obtain a classification of the concise cyclic presentations where the components of the corresponding star graph are generalized polygons. The classification reveals that both connected and disconnected star graphs are possible and that only generalized triangles (i.e. incidence graphs of projective planes) and regular complete bipartite graphs arise as the components. We list the presentations that arise in the Euclidean case and show that at most two of the corresponding groups are not SQ-universal (one of which is not SQ-universal, the other is unresolved). We obtain results that show that many of the SQ-universal groups are large

The SQ universality of some small cancellation groups

PhDA group G is a small cancellation group if, roughly, it has a presentation G= <A; R with the property that for any pair r, s of elemets of R either r=s1 or there is very little free cancellation in forming the product rs. The classical example of such a group is the fundamental group of a closed orientable 2-manifold of genus k which has a presentation k G=< al, bl, ..., ak, bk; 'TT \ai, bi' i=1 A countable group G is SQ-universal if every countable group can be embedded =in some quotient of G. The obvious example of SQ-universal group is the free group of rank 0. This work is a study of the SQ-universality of some small cancellation groups. A theory of diagrams is investigated in some detail- to be used as a tool in this study. The main achievement in this work is the following two results: (1) With few exceptions a small cancellation group contains nonabelian free subgroups. ( The emphasis here is on the nature of the free generators. ) (2) A characterization of the S Q-universality of some small cancellation groups