Grafting rays fellow travel Teichmuller geodesics

Details Grafting rays fellow travel Teichmuller geodesics Grafting rays fellow travel Teichmuller geodesics

2010-03-03

arxiv.org/abs/1003.0711v3

Mathematics

Geometric Topology

39 pages, 5 figures. v3: Minor corrections. v2: Main theorem improved using recent work of Rafi (arXiv:1011.6004), now compare

Scientific paper

Given a measured geodesic lamination on a hyperbolic surface, grafting the surface along multiples of the lamination defines a path in Teichmuller space, called the grafting ray. We show that every grafting ray, after reparametrization, is a Teichmuller quasi-geodesic and stays in a bounded neighborhood of a Teichmuller geodesic. As part of our approach, we show that grafting rays have controlled dependence on the starting point. That is, for any measured geodesic lamination Lambda, the map of Teichmuller space which is defined by grafting along Lambda is L-Lipschitz with respect to the Teichmuller metric, where L is a universal constant. This Lipschitz property follows from an extension of grafting to an open neighborhood of Teichmuller space in the space of quasi-Fuchsian groups.

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