Bieri-Strebel Groups With Irrational Slopes

We consider a group of continuous piecewise-linear 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 tree-pair 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 tree-pair 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.