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Kleinian groups and the complex of curves

Yair N Minsky

Geometry & Topology 4 (2000) 117–148

arXiv: math.GT/9907070


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.

Kleinian group, ending lamination, complex of curves, pleated surface, bounded geometry, injectivity radius
Mathematical Subject Classification 2000
Primary: 30F40
Secondary: 57M50
Forward citations
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
Yair N Minsky
Department of Mathematics
SUNY at Stony Brook
Stony Brook
New York 11794