The Effective Ehrenpreis Conjecture

TL;DR

This paper establishes an explicit polynomial bound on the degrees of finite covers needed to approximate two hyperbolic Riemann surfaces within any given Teichmüller distance, refining the Ehrenpreis Conjecture.

Contribution

It proves a uniform polynomial bound on the degrees of covers for the Ehrenpreis Conjecture, showing the bound is optimal for certain arithmetic surfaces.

Findings

Existence of a constant k such that degrees are less than epsilon^{-k}

Bound is optimal for arithmetic Riemann surfaces with same invariant trace field

Provides explicit quantitative bounds improving previous qualitative results

Abstract

Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $\epsilon>0$ there are finite covers $M_\epsilon \to M$, and $N_\epsilon \to N$, such that the Teichmuller distance (in the suitable moduli space) between $M_\epsilon$ and $N_\epsilon$ is less than $\epsilon$. It is natural to ask how large the degrees of these coverings need to be to achieve that the distance between $M_\epsilon$ and $N_\epsilon$ is less than $\epsilon$. In this paper we show that there exists a constant $k>0$, depending only on $M$ and $N$, so that the covers $M_\epsilon \to M$, and $N_\epsilon \to N$, can be chosen to have the degrees less than $\epsilon^{-k}$. We show that this bound is optimal by considering the case when $M$ and $N$ are arithmetic Riemann surfaces with the same invariant trace field which are not commensurable to each other.