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Let $\Gamma$ be a lattice in $G=\mathrm{SL}(2,\mathbb{C})$. We give aneffective equidistribution result with precise error terms for expandingtranslates of pieces of horospherical orbits in $\Gamma\backslash G$. Ourmethod of proof relies on the theory of unitary representations.

On the rate of equidistribution of expanding horospheres in finite-volume quotients of $\mathrm{SL}(2,\mathbb{C})$