Decision problems for finite and infinite presentations of groups and monoids

TL;DR

This survey explores how classical results on the Word Problem for finite group presentations extend to monoids and groups with possibly infinite presentations, introducing functions analogous to the Dehn function and invariance under Tietze transformations.

Contribution

It generalizes known results on the Word Problem to infinite presentations and introduces new functions and invariance concepts for decision problems.

Findings

Extension of Word Problem results to infinite presentations

Introduction of functions analogous to Dehn function for decision problems

Identification of Tietze transformations preserving these functions

Abstract

In this survey we show how well known results about the Word Problem for finite group presentations can be generalized to the Word Problem and other decision problems for non-necessarily finite monoid and group presentations. This is done by introducing functions playing the same role of the Dehn function for the given decision problem and by finding the Tietze transformations that leave this function invariant. This survey presents some original ideas and points of view.