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

TL;DR

This paper establishes polynomial effective density results for certain group orbits in arithmetic quotients of hyperbolic 3-space and products of hyperbolic planes, using advanced geometric and spectral methods.

Contribution

It provides the first polynomial error rate effective density theorems for orbits in these specific arithmetic quotients, combining Margulis functions, incidence geometry, and spectral gap techniques.

Findings

Proved polynomial error rate effective density theorems.

Applied Margulis functions and incidence geometry tools.

Utilized spectral gap properties of the ambient space.

Abstract

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