Polynomial effective density in quotients of ${\mathbb{H}}^3$ and ${\mathbb{H}}^2\times {\mathbb{H}}^2$

Abstract

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

76 pages