Polynomial effective equidistribution

Abstract

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\mathrm{SL}_2(\mathfrak{l})$ in arithmetic quotients of $\mathrm{SL}_2(\mathbb{C})$ and $\mathrm{SL}_2(\mathfrak{l})\times \mathrm{SL}_2(\mathfrak{l})$.The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.