Abstract
Let Γ<\PSL(2,\C) be a geometrically finite non-elementary discrete subgroup, and let its critical exponent δ be greater than 1. We use representation theory of \PSL(2,\C) to prove an effective bisector counting theorem for Γ, which allows counting the number of points of Γ 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.
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