On diagram groups over Fibonacci-like semigroup presentations and their generalizations

Abstract: We answer the question by Matt Brin on the structure of diagram groups over semigroup presentation $${\mathcal P}=\langle a,b,c\mid a=bc,b=ca,c=ab\rangle$$. In the talk on Oberwolfach workshop, Brin conjectured that the diagram group over $\mathcal P$ with base $a$ is isomorphic to the generalized Thompson's group $F_9$. We confirm this conjecture and consider some generalizations of this fact.