Geometry & Topology

Kleinian groups and the complex of curves

Yair N Minsky

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Abstract

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.

Article information

Source
Geom. Topol., Volume 4, Number 1 (2000), 117-148.

Dates
Received: 16 July 1999
Revised: 9 November 1999
Accepted: 20 February 2000
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883280

Digital Object Identifier
doi:10.2140/gt.2000.4.117

Mathematical Reviews number (MathSciNet)
MR2182094

Zentralblatt MATH identifier
0953.30027

Subjects
Primary: 30F40: Kleinian groups [See also 20H10]
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
Kleinian group ending lamination complex of curves pleated surface bounded geometry injectivity radius

Citation

Minsky, Yair N. Kleinian groups and the complex of curves. Geom. Topol. 4 (2000), no. 1, 117--148. doi:10.2140/gt.2000.4.117. https://projecteuclid.org/euclid.gt/1513883280


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