Groups with special presentations and star-graph $K_{3,3}$

Abstract: We consider a question of Edjvet and Vdovina concerning which groups defined by special presentations are large. For each integer $n \ge 3$, we construct an $n$-generator one-relator presentation whose star graph is the complete bipartite graph $K_{n,n}$; the resulting groups are large and hyperbolic. We also classify concise special presentations with star graph $K_{3,3}$, showing that they are one-relator presentations and that, up to Tietze equivalence, there are exactly twelve that define torsion-free groups. The torsion cases arise precisely as positive powers of the relators in the torsion-free cases, and define pairwise non-isomorphic groups that remain large and hyperbolic.