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

TL;DR

This paper proves that the diagram group over a specific Fibonacci-like semigroup presentation is isomorphic to a generalized Thompson's group, confirming a conjecture and exploring related generalizations.

Contribution

It confirms Brin's conjecture that the diagram group over a particular presentation is isomorphic to a generalized Thompson's group and investigates similar structures.

Findings

The diagram group over the given presentation is isomorphic to generalized Thompson's group F_9.

Confirmed Brin's conjecture about the structure of this diagram group.

Explored generalizations of the initial Fibonacci-like semigroup presentation.

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.