Average bending of convex pleated planes in hyperbolic three-space

Abstract

If P is a pleated plane in 3-dimensional hyperbolic space H 3 and α a geodesic in its intrinsic metric we define B(P,α), the average bending of P in the direction α. We show that if P is a convex pleated plane embedded in H 3 then B(P,α)≤K for some universal K. Furthermore if PΓ is a boundary component of the convex hull of a quasi-Fuchsian group Γ then B(PΓ,α)=B(Γ) almost everywhere, where B(Γ) is a constant times the length of the bending lamination βΓ of the pleated surface X Γ=PΓ/Γ. We use these to prove a number of results about quasi-Fuchsian groups including a universal bound on the Lipschitz constant for the map to infinity and a bound on the length of βΓ by a constant times the Euler characteristic of X Γ.