Effective bisector estimate with application to Apollonian circle packings

Let \Gamma<\PSL(2,\C) be a geometrically finite non-elementary discrete subgroup, and let its critical exponent \delta\ be greater than 1. We use representation theory of \PSL(2,\C) to prove an effective bisector counting theorem for \Gamma, which allows counting the number of points of \Gamma\ in general expanding regions in \PSL(2,\C) and provides an explicit error term. We apply this theorem to give power savings in the Apollonian circle packing problem and related counting problems.