**Bieri-Strebel Groups With Irrational Slopes.** / Winstone, Nicholas.

Research output: Thesis › Doctoral Thesis

Unpublished

**Bieri-Strebel Groups With Irrational Slopes.** / Winstone, Nicholas.

Research output: Thesis › Doctoral Thesis

Winstone, N 2022, 'Bieri-Strebel Groups With Irrational Slopes', Ph.D., Royal Holloway, University of London.

Winstone N. Bieri-Strebel Groups With Irrational Slopes. 2022. 190 p.

@phdthesis{c4f499b2ec7b436eb23c08e777326fec,

title = "Bieri-Strebel Groups With Irrational Slopes",

abstract = "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.",

keywords = "Groups, Pisot numbers, Graph Theory, trees (mathematics)",

author = "Nicholas Winstone",

year = "2022",

language = "English",

school = "Royal Holloway, University of London",

}

TY - THES

T1 - Bieri-Strebel Groups With Irrational Slopes

AU - Winstone, Nicholas

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Groups

KW - Pisot numbers

KW - Graph Theory

KW - trees (mathematics)

M3 - Doctoral Thesis

ER -