  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.
  We consider groups $S$ with presentations of deficiency 1 satisfying certain
technical conditions and construct balanced group presentations $\P_w$ indexed
by words $w$ in the generators of $S$. If $w=1$ in $S$ then $\P_w$ 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 $S$.
