An introduction to diagram groups
Anthony Genevois · Journal of Combinatorial Algebra · 2023
To every semigroup presentation \mathcal{P}= \langle \Sigma | \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 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 wreath product \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.