EQUIDISTRIBUTION AND COUNTING FOR ORBITS OF GEOMETRICALLY FINITE HYPERBOLIC GROUPS

Abstract

Let Q be a quadratic form of signature (n, 1), Γ a non-elementary discrete subgroup of G = SOQ(R) and w0 ∈ Rn+1 a non-zero vector with the orbit w0Γ discrete. We introduce the notion of the Γ-skinning size skΓ(w0) of w0, and we compute an asymptotic for-mula (as T → ∞) for the number of points in w0Γ of norm at most T, provided both the Bowen-Margulis-Sullivan measure mBMSΓ and the Γ-skinning size skΓ(w0) are finite. For Γ geometrically finite, it is known that |mBMSΓ | < ∞ by Sullivan and we give a criterion on the finiteness of skΓ(w0): skΓ(w0) < ∞ if and only if either δΓ> 1 or w0 is not exter-nally Γ-parabolic. We also prove new weighted equidistribution theorem for normal geodesic evolution of codimension one totally geodesic im-mersions. However the applicability of the basic result is not limited to geometrically finite groups We use these results to count circles in the circle packing P of an ideal triangle of the hyperbolic plane H2 made by repeatedly inscribing the largest circles into the triangular interstices. We obtain that the number of circles in P of hyperbolic curvature at most T is asymptotic to c · T 1.30568(8) where c = c(H2)> 0 is an absolute constant.