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

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.