UC Santa Barbara

The Andrews–Curtis conjecture, proposed in 1965, asks whether any balanced presentation of the trivial group can be transformed to the trivial presentation using a specific set of moves. This thesis investigates both the standard conjecture and its stable variant from algorithmic and topological perspectives.We develop a state-of-the-art algorithm for trivializing group presentations and apply it to resolve several open problems. We prove that all but two presentations in the Akbulut–Kirby series can be reduced in length, and we resolve various potential counterexamples in the Miller–Schupp series, including three infinite subfamilies.The stable Andrews–Curtis conjecture allows additional stabilization moves and is equivalent to a topological statement: any contractible 2-dimensional polyhedron can be 3-deformed to a point. We investigate this formulation through the systematic study of fake surfaces, which are generic 2-polyhedra. We present a complete classification of acyclic cellular fake surfaces up to complexity 4, and classify complexity 5 surfaces without small disks. This classification reveals 2, 17, 238, and 4618 distinct surfaces of complexities 1 through 4, respectively.Using this classification, we establish an inductive scheme for proving new cases of the stable Andrews–Curtis and Zeeman conjectures and verify it holds up to complexity 5. As consequences, we prove the contractibility conjecture for all acyclic fake surfaces of complexity 4 and verify the embedded disk conjecture up to complexity 5.