The existence of quasiconformal homeomorphism between planes with countable marked points

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 $\mathbb{C}\setminus \mathbb{Z}$.

13 pages, 4 figures. arXiv admin note: text overlap with arXiv:1405.0340