Abstract
We consider a group of continuous piecewiselinear homeomorphisms of the unit interval for which every slope has gradient which is a power of some quadratic integer. We aim to determine for which choices of this quadratic integer we can represent every such homeomorphism as a treepair diagram. We have been able to successfully identify necessary and sufficient conditions on these quadratic integers, and manage to extend some of these results to algebraic integers of higher degree.
For groups such that the quadratic integer satisfies these conditions, and hence treepair representation exist for each homeomorphism, we find explicit presentations, notable for containing only two families of generators and two families of relations. We have also determined presentations for the abelianisation of such groups, and furthermore found some abelianisations of these groups with arbitrarily high torsion.
For groups such that the quadratic integer satisfies these conditions, and hence treepair representation exist for each homeomorphism, we find explicit presentations, notable for containing only two families of generators and two families of relations. We have also determined presentations for the abelianisation of such groups, and furthermore found some abelianisations of these groups with arbitrarily high torsion.
Original language  English 

Qualification  Ph.D. 
Awarding Institution 

Supervisors/Advisors 

Award date  1 Jun 2022 
Publication status  Unpublished  2022 
Keywords
 Groups
 Pisot numbers
 Graph Theory
 trees (mathematics)