# Finitely-presented groups

Lead Research Organisation:
University of St Andrews

Department Name: Mathematics and Statistics

### Abstract

One of the main ways to represent an infinite group using only a finite amount of data is via a presentation. We list a set of generators for the group, and a set of rules called relators that define the multiplication.

Whilst this gives a very compact representation, unfortunately it is not possible in general to answer certain questions about the group, given only the presentation. For example, there is no algorithm to determine whether a group given by a finite presentation is finite or infinite.

One class of finitely-presented groups for which these problems are decidable is the class of hyperbolic groups. These have a geometric definition, and naturally act on a hyperbolic metric space. One remarkable fact about hyperbolic groups is that (using many different standard definitions of a random group) any random group is hyperbolic with probability tending to 1 as the size of the presentation grows. Unfortunately, it is undecidable in general whether or not a given group is hyperbolic.

Before the definition of hyperbolic groups was given by Gromov in the 1980s, many researchers had studied small cancellation presentations, which are presentations with especially nice "overlap" properties between the rules defining the multiplication.

We will look at a variety of decision problems for hyperbolic groups, including the word and conjugacy problem, seeking to understand how techniques from small cancellation theory can be used to show that a given presentation is hyperbolic, and to find effective solutions to various decision problems. We will also look at developing new models of random presentations, to see whether these produce a wider class of groups than just the hyperbolic groups.

Whilst this gives a very compact representation, unfortunately it is not possible in general to answer certain questions about the group, given only the presentation. For example, there is no algorithm to determine whether a group given by a finite presentation is finite or infinite.

One class of finitely-presented groups for which these problems are decidable is the class of hyperbolic groups. These have a geometric definition, and naturally act on a hyperbolic metric space. One remarkable fact about hyperbolic groups is that (using many different standard definitions of a random group) any random group is hyperbolic with probability tending to 1 as the size of the presentation grows. Unfortunately, it is undecidable in general whether or not a given group is hyperbolic.

Before the definition of hyperbolic groups was given by Gromov in the 1980s, many researchers had studied small cancellation presentations, which are presentations with especially nice "overlap" properties between the rules defining the multiplication.

We will look at a variety of decision problems for hyperbolic groups, including the word and conjugacy problem, seeking to understand how techniques from small cancellation theory can be used to show that a given presentation is hyperbolic, and to find effective solutions to various decision problems. We will also look at developing new models of random presentations, to see whether these produce a wider class of groups than just the hyperbolic groups.

## People |
## ORCID iD |

Colva Mary Roney-Dougal (Primary Supervisor) | |

Simon Jurina (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/R513337/1 | 01/10/2018 | 30/09/2023 | |||

2095926 | Studentship | EP/R513337/1 | 01/10/2018 | 31/03/2022 | Simon Jurina |