The Grushin plane and quasiconformal Jacobians

Abstract

We construct a quasiconformal map from the Grushin plane to the Euclidean plane. Then, we generalize the Grushin plane slightly and explain how the Grushin plane can serve as an intermediary in dealing with quasiconformal maps on Euclidean spaces. In particular, we construct a family of quasiconformal embeddings of the Euclidean plane into larger Euclidean spaces whose Jacobians fail to be locally integrable on a line.