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.