Geometry & Topology, Vol. 4 (2000) Paper no. 3, pages 117--148.

Kleinian groups and the complex of curves

Yair N Minsky

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.

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

AMS subject classification. Primary: 30F40. Secondary: 57M50.

DOI: 10.2140/gt.2000.4.117

E-print: arXiv:math.GT/9907070

Submitted to GT on 16 July 1999. (Revised 9 November 1999.) Paper accepted 20 February 2000. Paper published 29 February 2000.

Notes on file formats

Yair N Minsky
Department of Mathematics, SUNY at Stony Brook
Stony Brook, NY 11794, USA

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