Kleinian groups and the complex of curves

Details Kleinian groups and the complex of curves Kleinian groups and the complex of curves

1999-07-12

arxiv.org/abs/math/9907070v2

Geom. Topol. 4 (2000), 117-148

Mathematics

Geometric Topology

32 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper3.abs.html

Scientific paper

We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have `bounded geometry' (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed using the ending invariants of the group and the complex of curves. These results are directly analogous to those obtained in the case of punctured-torus surface groups. In that setting the ending invariants are points in the closed unit disk and the coefficients are closely related to classical continued-fraction coefficients. The estimates obtained play an essential role in the solution of Thurston's ending lamination conjecture in that case.

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