The existence of quasiconformal homeomorphism between planes with countable marked points

TL;DR

This paper investigates conditions under which infinitely punctured planes, specifically those with countable marked points, are quasiconformally equivalent to the complex plane minus integers, providing criteria and characterizations.

Contribution

It offers new criteria and characterizations for when certain infinitely punctured planes are quasiconformally equivalent to the plane minus integers.

Findings

Criteria for quasiconformal equivalence of punctured planes

Characterization of subsets of real line with quasiconformal complements

Conditions for equivalence to the complex plane minus integers

Abstract

We consider quasiconformal deformations of $\mathbb{C}\setminus\mathbb{Z}$. We give some criteria for infinitely often punctured planes to be quasiconformally equivalent to $\mathbb{C}\setminus\mathbb{Z}$. In particular, we characterize the closed subsets of $\mathbb{R}$ whose compliments are quasiconformally equivalent to $\mathbb{C}\setminus\mathbb{Z}$.