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Kleinian groups and the complex of curves
Yair N Minsky
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Geometry & Topology 4 (2000)
117–148
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Abstract
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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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Keywords
Kleinian group, ending lamination,
complex of curves, pleated surface, bounded geometry,
injectivity radius
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Mathematical Subject Classification
Primary: 30F40
Secondary: 57M50
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Publication
Received: 16 July 1999
Revised: 9 November 1999
Accepted: 20 February 2000
Published: 29 February 2000
Proposed: David Gabai
Seconded: Jean-Pierre Otal, Walter Neumann
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