Dehn Functions, the Word Problem, and the Bounded Word Problem For Decidable Group Presentations

TL;DR

This paper constructs and analyzes finitely generated decidable group presentations with specific properties related to the word problem, bounded word problem, and Dehn function, answering an open question in the field.

Contribution

It provides explicit examples of such presentations, proves the non-existence of others, and generalizes existing machinery to simulate complex computational models.

Findings

Constructed minimal finitely generated decidable presentations with specific properties.

Proved non-existence of certain combinations of properties.

Generalized machinery to simulate oracle Turing machines in group presentations.

Abstract

We construct examples of finitely generated decidable group presentations that satisfy certain combinations of solvability for the word problem, solvability for the bounded word problem, and computablity for the Dehn function. We prove that no finitely generated decidable presentations exist satisfying the combinations for which we do not provide examples. The presentations we construct are also minimal. These constructions answer an open question asked by R.I. Grigorchuk and S.V. Ivanov. Our approach uses machinery developed by Birget, Ol'shanskii, Rips, and Sapir for constructing finite group presentations that simulate Turing machines. We generalize this machinery to construct finitely generated decidable group presentations that simulate computing objects similar to oracle Turing machines.