On quasiconformal equivalence between certain infinitely often punctured planes

TL;DR

This paper investigates conditions under which certain infinitely punctured planes are quasiconformally equivalent, establishing criteria for tameness and demonstrating that the lattice points in the plane do not satisfy these conditions.

Contribution

The paper provides new criteria for tameness of discrete sets and proves that the integer lattice is not tame, advancing understanding of quasiconformal equivalences in complex analysis.

Findings

Several criteria for a set to be tame are established.

The integer lattice $\mathbb{Z}+i\mathbb{Z}$ is shown not to be tame.

The results clarify which punctured planes are quasiconformally equivalent.

Abstract

A closed discrete subset $A\subset \mathbb{C}$ is called tame if $\mathbb{C}\setminus A$ is quasiconformally equivalent to $\mathbb{C}\setminus \mathbb{Z}$. By giving several criteria for $A$ to be tame, we shall show that $\mathbb{Z}+i\mathbb{Z}$ is not tame.