Mathematical Research Letters

Volume 24 (2017)

Number 4

Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces

Pages: 1177 – 1203



Matthew Romney (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.)

Vyron Vellis (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)


We show that, for all $\alpha \geq 0$, the generalized Grushin plane $\mathbb{G}_{\alpha}$ is bi-Lipschitz homeomorphic to a 2-dimensional quasiplane in the Euclidean space $\mathbb{R}^{\lfloor \alpha \rfloor +2}$, where $\lfloor \alpha \rfloor$ is the integer part of $\alpha$. The target dimension is sharp. This generalizes a recent result of Wu [J.-M.Wu, “Bilipschitz embedding of Grushin plane in $\mathbb{R}^3$”, Ann. Sc. Norm. Super. Pisa Cl. Sci. 14 (2015), no. 2, 633–644].


generalized Grushin plane, quasiplane, bi-Lipschitz embedding

2010 Mathematics Subject Classification

Primary 53C17. Secondary 30L05, 30L10.

The second author was supported by the Academy of Finland project 257482.

Received 23 March 2015

Accepted 12 February 2016

Published 9 November 2017