Relative hyperbolicity of graphs of free groups with cyclic edge groups

• Main Author: Richer, Émilie.
• Format: Others
• Language: en
• Published: McGill University 2006
• Subjects: Hyperboloid.; Free groups.
• Online Access: http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=101170

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.