Explore a 54-minute lecture on division algorithms, group rings, and negative curvature in low-dimensional topology. Delve into the fundamental problem of understanding 2-complexes with a given fundamental group G. Learn how this challenge can be approached using a division algorithm in the group ring of G, and discover the conditions under which such an algorithm becomes available. Examine additional applications, including methods for establishing lower bounds on the complexity of cell-decompositions in hyperbolic manifolds. This talk, part of the Workshop on Groups around 3-Manifolds, offers valuable insights into the intersection of algebra, topology, and geometry.