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

TL;DR

This paper proves that groups satisfying certain geometric small-cancellation conditions, with additional restrictions, are bi-automatic, thus addressing gaps in previous algebraic small-cancellation group proofs.

Contribution

It establishes that geometric C(4) and T(4) small-cancellation conditions imply bi-automaticity under specific additional restrictions, closing gaps in earlier algebraic proofs.

Findings

Geometric small-cancellation conditions imply bi-automaticity with restrictions

Additional edge label restrictions are necessary for the implication

Barycentric subdivision method is used in the proof

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.