We prove that any finitely generated group which splits as a graph of free groups with cyclic edge groups is hyperbolic relative to certain finitely generated subgroups, known as the peripheral subgroups. Each peripheral subgroup splits as a graph of cyclic groups. Any graph of free groups with cyclic edge groups is the fundamental group of a graph of spaces X where vertex spaces are graphs, edge spaces are cylinders and attaching maps are immersions. We approach our theorem geometrically using this graph of spaces. / We apply a "coning-off" process to peripheral subgroups of the universal cover X̃ → X obtaining a space Cone(X̃) in order to prove that Cone (X̃) has a linear isoperimetric function and hence satisfies weak relative hyperbolicity with respect to peripheral subgroups. / We then use a recent characterisation of relative hyperbolicity presented by D.V. Osin to serve as a bridge between our linear isoperimetric function for Cone(X̃) and a complete proof of relative hyperbolicity. This characterisation allows us to utilise geometric properties of X in order to show that pi1( X) has a linear relative isoperimetric function. This property is known to be equivalent to relative hyperbolicity. / Keywords. Relative hyperbolicity; Graphs of free groups with cyclic edge groups, Relative isoperimetric function, Weak relative hyperbolicity.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:QMM.101170 |
| Date | January 2006 |
| Creators | Richer, Émilie. |
| Publisher | McGill University |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Format | application/pdf |
| Coverage | Master of Science (Department of Mathematics and Statistics.) |
| Rights | © Émilie Richer, 2006 |
| Relation | alephsysno: 002600043, proquestno: AAIMR32779, Theses scanned by UMI/ProQuest. |
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