There is long standing question of whether a group with a finite K(G,1) that does not contain Baumslag–Solitar subgroups must be hyperbolic. The presentation complex of a group presentation with one defining relator is a finite K(G,1). Louder and Wilton showed that if the defining relator has imprimitivity rank greater than 2 then the group does not contain Baumslag–Solitar subgroups, so they conjecture that such one-relator groups are hyperbolic. With Hoffmann, we verified this conjecture computationally for all one-relator groups in which the length of the relator is at most 17.
In this talk I'll introduce hyperbolic, one-relator, and Baumslag–Solitar groups, and talk about how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams. I'll also talk about primitivity of elements in a free group and how to compute imprimitivity rank using Stallings subgraphs.