The complexity of balanced presentations and the Andrews-Curtis conjecture

<p>Motivated by problems in topology, we explore the complexity of balanced group presentations. We obtain large lower bounds on the complexity of Andrews-Curtis trivialisations, beginning in rank 4. Our results are based on a new understanding of how Dehn functions of groups behave under certain kinds of push-outs.</p> <br/> <p>We consider groups S with presentations of deficiency 1 satisfying certain technical conditions and construct balanced group presentations Pw indexed by words w in the generators of S. If w = 1 in S then Pw is Andrews-Curtis trivialisable and the number of Andrews-Curtis moves required to trivialise it can be bounded above and below in terms of how hard it is to prove that w = 1 in S.</p>