Kleinian groups and the complex of curves

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

Mathematical Subject Classification 2000

Primary: 30F40

Secondary: 57M50

References

Forward citations

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

Authors

Yair N Minsky
Department of Mathematics SUNY at Stony Brook Stony Brook New York 11794 USA

Volume 4, issue 1 (2000)

Yair N Minsky