Geometry of graphs of discs in a handlebody

Details Geometry of graphs of discs in a handlebody Geometry of graphs of discs in a handlebody

2011-01-10

arxiv.org/abs/1101.1843v2

Mathematics

Geometric Topology

33 pages. Completely rewritten. Substantial correction due to an overlooked case in a surgery argument. An independent proof o

Scientific paper

For a handlebody H of genus n>1 we investigate three different graphs of discs and show that they are hyperbolic. One of the graphs is the electrified disc graph whose vertices are isotopy classes of essential discs in H. Two discs are connected by an edge of length one if there is a simple closed curve disjoint from both. We use hyperbolicity of the electrified disc graph to give an alternative proof of hyperbolicity of the disc graph. We also show that there is a coarse surjective Lipschitz projection of the electrified disc graph on the free factor graph of the free group with n generators.

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