An introduction to diagram groups

TL;DR

This paper introduces diagram groups, a class of groups associated with semigroup presentations, highlighting their properties, examples, and significance in geometric group theory.

Contribution

It provides a comprehensive survey of diagram groups, summarizing known results, examples, and their relevance in the study of group theory.

Findings

Includes Thompson's group F as a diagram group

Shows diagram groups encompass lamplighter and braid groups

Highlights the role of diagram groups in geometric group theory

Abstract

To every semigroup presentation $\mathcal{P}= \langle \Sigma \mid \mathcal{R} \rangle$ and every baseword $w \in \Sigma^+$ can be associated a diagram group $D(\mathcal{P},w)$, defined as the fundamental group of the so-called Squier complex $S(\mathcal{P},w)$. Roughly speaking, $D(\mathcal{P},w)$ encodes the lack of asphericity of $\mathcal{P}$. Examples of diagram groups include Thompson's group $F$, the lamplighter group $\mathbb{Z} \wr \mathbb{Z}$, the pure planar braid groups, and various right-angled Artin groups. This survey aims at summarising what is known about the family of diagram groups.