Algebraic C(4) and T(4) groups are bi-automatic

Abstract: S. Gersten and H. Short have proved that if a group has a presentation which satisfies the algebraic C(4) and T(4) small-cancellation condition then the group is automatic. Their proof contains a gap which we aim to close. To do that we distinguish between algebraic small-cancellation conditions and geometric small cancellation conditions (which are conditions on the van Kampen diagrams). We show that, under certain additional requirements, geometric C(4) and T(4) small-cancellation conditions imply bi-automaticity. The additional requirements include a restriction on the labels of edges in minimal van Kampen diagrams. This, together with the so-called barycentric sub-division method proves the theorem.