An introduction to diagram groups

To every semigroup presentation $\mathcal{P}=⟨\mathrm{\Sigma }\mid \mathcal{R}⟩$ and every baseword $w\in {\mathrm{\Sigma }}^{+}$ 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.