Morse theory for group presentations (Q6567144)

The Andrews-Curtis conjecture states that any presentation of the trivial group can be transformed into the empty presentation by a finite sequence of \(Q^{**}\)-transformations. In this article, a method is presented which combines topological and combinatorial tools that allows the computational exploration of presentations which are \(Q^{**}\)-equivalent to a given one without the need of exhibiting the actual list of transformations. This alternative technique to find \(Q^{**}\)-transformations is based on a refinement of discrete Morse theory. With its help some well-known potential counterexamples to the Andrews-Curtis conjecture can be easily \(Q^{**}\)-trivialized. Moreover, this theory is applied to presentations of non-trivial groups, proving that some potential counterexamples to the generalized Andrews-Curtis conjecture do satisfy it.