Minimal presentations of groups
This thesis proposes two new families of counter examples to the celebrated relation gap problem: firstly certain finite-index subgroups of right-angled Artin groups, and secondly groups with torsion arising as free products of HNN extensions of finite abelian groups. We prove a number of results relating the size of minimal presentations of the Artin subgroups to topological conditions on the associated flag complexes. We construct explicit presentations of these groups that provide considerable insight into when and why large families of relations are needed to present these groups, and cumulative evidence that these subgroups are extremely strong candidates for groups with relation gaps. For the examples with torsion, we prove that our groups have presentations whose associated relation modules can be generated by one fewer element than one expects. We prove that any presentation with this number of relations must be of a peculiar form. We also show that a criterion ofM. Lustig provides a non-trivial obstruction to the existence of relation gaps, but that for the presentations we are studying, this obstruction never arises. Finally, we give an explicit description of a finite 3-complex that is a counterexample to the D(2) onjecture if our groups have relation gaps.